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TRIANGULATION 

Applied to Sheet 
Metal Pattern Cutting 



A Comprehensive Treatise for Cutters, Draftsmen, 
Foremen and Students; Progressing from the 
simplest phases of the subject to the most complex 
problems employed in the development of Sheet 
Metal Patterns; With practical solutions of nu- 
merous problems of frequent occurrence in sheet 
metal shops. 

By F. S. KIDDER 



Illustrated by means of 1 24 engravings in 
line and half-tone, including many repro- 
ductions of photographs of sheet metal 
models, made expressly for this work. 



NEW YORK 

The Sheet Metal Publication Company 

19 17 



,V\5 



Copyright, 1917 

THE SHEET METAL PUBLICATION COMPANY 

NEW YORK 



\1- 



y 

MAY 29 S9I7 



'CI.A467201 



CONTENTS 

CHAPTER. PAGE. 

J. Elementary Principles 3 

■II. A Simple Transitional Fitting 12 

IIT. The Oblique Cone 19 

IV. A Transitional Fitting From Rectangular to 

Round Which Makes an Offset. 25 

V. A Twisted Transitional Fitting 33 

VI. The Pattern for the Frustum of an Oblique Cone 41 

VII. A Transitional Fitting From Oblong to Round. . 50 

VIII. A Two-Pronged Fitting Which Can Be Made in 

One Piece 58 

IX. Some Principles of Orthographic Projection as 

Applied to Triangulation 67 

X. The Representation of Objects Upon the Vertical, 
Horizontal, Profile and Oblique .Supplemen- 
ary Planes of Projection 74 

XI. The Pattern for a Fitting Whose Ends Are Xot 

in Parallel Planes 81 

XII. The Pattern for a Fitting Whose Ends Are Not 

in Parallel Planes, Second Demonstration. ... 90 

XIII. A Transitional Elbow From Round to Rectangu- 

lar 96 

XIV. A Transitional Offset From Round to Rectan- 

gular 106 

XV. A Three-Pieced Tapering Elbow 113 

XVI. The Ship's Ventilator. .'..,- 124 

XVII. ( )n the Tapering Elbow to be made in Any Num- 
ber of Pieces 133 

XVIII. A Transitional Elbow in Rectangular Pipe 141 

XIX. A Transitional Elbow From Round to Elliptical 148 



CHAPTER. PAGE. 

XX. The Helical Elbow : 160 

XXI. When It Is Required That a Round Pipe Should 

Join the Frustum of an Oblique Cone 172 

XXII. A Branched Fitting Commonly Known as 

"Breeches" 186 

XXIII. A Simple Two-Pronged Fitting 198 

XXIV. A Two-Pronged Fitting Whose Prongs Are 

Unequal 207 

XXV. On the Two-Pronged Fitting When It Is 
Required That the Prongs Radiate at a Given 

Angle to the Main Stem 217 

XXVI. On a Fitting With Any Number of Prongs 224 

XXVII. ' The Right or Scalene Cone Considered in Secur- 
ing the Patterns for a Branched Fitting 232 

Concluding Remarks 251 

Glossary 255 

Index 263 



Preface 



The great and increasing demand for irregular forms 
to be made from sheet metal has made Triangulation an 
important factor in Sheet Metal Pattern Development. 
This has induced the writer to lay before the sheet metal 
worker, a work designed for the purpose of enabling him 
to acquire a thorough understanding of this branch of 
Pattern Cutting. 

Triangulation has in many instances been a subject of 
more or less mystery. However, from a study of Geo- 
metrical works, we conclude that its secrets have long 
been known. 

Few, if any writers upon Sheet Metal Pattern Devel- 
opment have seen fit to interpret it in a manner which 
affords the average worker an opportunity of grasping 
its underlying principles. The universal practise of 
most writers upon this subject has been to lay before 
the student worked out examples to be copied, little, 
or no attempt being made to convey an understanding- 
of the principles employed, which is of the utmost im- 
portance. 

No amount of time devoted to copying Chinese 
characters would enable one to understand them, nor 
will any amount of time consumed in copying pattern 
demonstrations enable one to understand the use and 
purpose of lines there found. 

In all examples of pattern development lines are pre- 
sumed to be upon the surface of the object. Upon de- 
termining the lengths of said lines, and the distances 



2 PREFACE 

they are from each other, we are enabled to place them 
upon the plane of development in their proper lengths 
and relative positions, thereby securing points through 
which lines are traced which represent the boundaries 
of the required pattern. 

There is a great sameness in the principles and methods 
which may be applied to many examples. In other 
words, if we grasp the reason for, and the use of each 
and every line in one problem, we are prepared to use 
those same principles and methods for all. 

Forms which must be treated by Triangulation are 
such that the rectilinear elements of their surfaces are 
neither parallel or convergent lines. Therefore to de- 
termine their lengths we must assume a supplementary 
plane for each, or employ the right angled triangle. To 
secure their relative positions, we must presume that the 
surface of the object is divided into triangles. 

Some idea of Orthographic Projection will be of 
service to the one who aspires to become proficient in 
this branch of pattern development, although the solving 
of a great number of the more common problems is but 
a simple operation. 

This work is submitted with every confidence that if 
attention is devoted to the subject matter enclosed, one 
will be enabled to more clearly understand the principles 
involved in Triangulation as Applied to Sheet Metal Pat- 
tern Development. 

F. S. K. 



TRIANGULATION 



CHAPTER I. 

Elementary Principles. 

Triangulation is a term which has in recent years been 
applied to certain operations in Sheet Metal Pattern De- 
velopment, although said operations have long been ex- 
plained in works upon Descriptive Geometry, where is 
found the declaration that the true length of a right line 
in space may always be found in the hypothenuse of a 
right angled triangle, whose base is equal in length to 
the horizontal projection of the line, and whose perpen- 
dicular is equal to the difference in length of the vertical 
projectors from the extremities of that line. 

Triangulation as applied to sheet metal pattern de- 
velopment, is the act or process of dividing into triangles, 
also the results thus secured; specifically, the laying out 
and accurate measurement of a network of triangles 
presumed to be upon the surface of the object, and shown 
upon its geometrical representation which has been cor- 
rectly delineated. 

Triangles with which we deal are considered as plane 
triangles, although not strictly so, since a plane triangle 
is presumed to lie in one plane, and is bounded by three 
right lines. A triangle which is presumed to be a por- 
tion of the curved surface of an object will not lie in one 
plane. In many instances one side at least of said tri- 
angle is not a right line but a curved one. Thus many 
triangles involved in triangulation as applied to sheet 



TRTANGULATION 



metal pattern development are mixtilinear triangles. 
However, the magnitude of the variation is so small, 
that it may consistently be considered as a negligible 
quantity. 

To Draw a Triangle. 

As an aid in securing a clear conception of the most 
elementary principles involved, we may for the moment 
presume that we have given us three, four and five 
inches as the lengths of three sides of a triangle. 

By the use of our compasses and straight edge, we are 
enabled to draw such a triangle by first drawing a line 
three inches long, as illustrated at a b, Fig. 1. 





Fig. 1. Illustrating a 
method of drawing a tri- 
angle to given dimen- 
sions. 



Fig. 2. Illustrating a 
method of drawing a 
parallelogram when the 
length of its diagonal is 
known. 



With compasses set to a span of four inches, place one 
point at a, and describe the arc c d. Since the length of 
one side of the required triangle is four inches, the vertex 
of one angle must lie in the arc c d, and as the third 
side is required to have a length of five inches, the com- 
passes may be adjusted to a span of five inches, and with 
point b as center, we may describe the small arc as at c, 



ELEMENTA RY PR] NfCIPLES 5 

thus locating the vertex of the third angle in its correct 
relative position. Lines may be drawn connecting points 
as shown in Fig. 1, thus forming the three sides of the 
required triangle. Here a b is the base, a c the perpen- 
dicular, and b c the hypothenuse. 

To Draw a Parallelogram. 

To draw a figure which is known as a parallelogram, 
to given dimensions, the magnitude of at least one angle, 
or the length of its diagonal, must be known. 

As for example, we have given us the lengths of two 
parallel sides as four inches, and the distance between 
the extremities of these lines as three inches. We draw 
a line whose length is four inches, as illustrated at a b, 
Fig. 2. 

Since the extremities of the line forming the opposite 
side are to be three inches from points a and b, we may 
adjust our compasses to a span of three inches, and with 
points a and b as centers, describe arcs as shown at c 
and d. As these arcs have been drawn with a radius of 
three inches, every point of which they are composed 
must be three inches distant from their respective centers 
a and b. Therefore to locate the line which forms the 
second four inch side, the magnitude of at least one 
angle, or as above stated, the length of the diagonal 
must be known. Presuming this to be five inches, we 
adjust our compasses to a span of five inches, and with 
point a or b as center, describe a small arc intersecting 
the first as at d, thereby locating the vertex of the angle 
as at d, in its correct relative position. Since the side 
c d is known to be four inches long, point c is located as 
shown; lines are now drawn to complete the required 
figure. 

Thus as will be noted, the parallelogram has been 



TRIANGULATION 



drawn by knowing the lengths of three sides of one of 
the two triangles of which it is composed. 

Some Suggestions. 
The student is advised to cut from sheet metal or 
cardboard two pieces, the forms of which are shown at 
Figs. 3 and 4. These forms may be looked upon as the 
forms of the top and base of an object which transforms 
from square to octagon, the square in this example 
being considered as the base. 





Fig. 3. Illustrating a 
form to be cut from 
sheet metal or card- 
board. 



Fig. 4. Illustrating a 
form to be cut from 
sheet metal or card- 
board. 



Something for a center support as a block of wood 
whose ends have been made parallel, may be secured, 
and of a length suitable for the vertical hight of an 
object whose size of base and top have previously been 
established in the pieces spoken of, and shown in Figs. 
3 and 4. These pieces may be fastened to the block of 
wood, and so arranged that lines a b, and c d, as shown 
at Figs. 3 and 4, will be parallel; with the center of the 
top directly above the center of the base, as illustrated 
at Fig. 5. 

We now have a form about which a flexible but non- 
elastic material (paper) may be formed, which if 
marked or trimmed at top and base, will when removed, 



EL EM KNTAR V TRINC 1 1 ' I . ES 



supply a pattern for an object whose dimensions have 
been established in said form, the surface of which may 
be looked upon as being composed of triangles. The 
lengths of sides of the square and octagon furnish the 
length of one side of each triangle. The lengths of the 
remaining two sides of each triangle in this example, 
will be foand in the distances points a and b of the top 
are from point d of the base, or, in the true lengths of 
lines a d and b d, Fig. 6. 





Fig. 5. Illustrating the rel- 
ative positions the two 
pieces of sheet metal or 
cardboard should occupy. 



Fig. 6. Illustrating trian- 
gles from which the true 
lengths of lines may be se- 
cured. 



It may be here remarked that in this example, it is 
only necessary to determine the lengths of two lines 
which represent the sides of one triangle, since there are 
in reality, but two lengths of side in the triangles form- 
ing the whole surface of the pattern, as will be herein- 
after shown. If lines be dropped from points a and b, 
as shown at a e, and b f, Fig. 6, which are perpendicular 
to the top and base, lines may be drawn as c d, and f d. 
YVe then have in the true lengths of lines a c and b f, 
the perpendiculars of right angled triangles, the bases 
of which are c d and f d, when, as is clearlv shown by 
Fig. 6, the true length of each line is found in the 
hvpothenuse of its respective triangle. 



8 TRIANGULATION 

From what has been stated above, the student will 
note that with the form previously constructed as shown 
at Fig. 5, he could drop imaginary lines as illustrated at 
Fig. 7, from the vertex of each angle of the octagon to 
the base, as in points g h i j k o n and in, and if lines be 
drawn upon the base to connect these points, he will have 
duplicated the form as previously established in the top. 





Fig. 7. Illustrating prin- 
ciples by which a plan is 
obtained. 



Fig. 8. A plan of the ob- 
ject. 



Thus, as will be noted, the same results would have been 
secured, had we constructed a diagram as shown at 



Fig. 8. 



A Plan. 



This diagram consists of the square abed, which is 
the exact form and size of the base, and is known as a 
plan of that portion of the object. The octagon e f g h 
i m k j, is the exact form and size of the top, and is 
known as a plan of that portion. 

Since in orthographic projection, the intersection of 
two planes demands a line, we draw lines as shown at 
e c, j c,k c,k d, etc., Fig. 8, which represent in plan the 
sides of triangles of which the surface of the object is 
composed. 



ELEMENTARY PRINCIPLES 9 

As the sides of the square and octagon supply the 
true lengths of one side of each triangle, we have simply 
to determine the true lengths of sides represented in lines 
k d, m d, etc., to furnish all measurements necessary for 
completing a pattern. It has been previously shown 
that each of these lengths may be found in the 
hypothenuse of a right angled triangle, whose base is 
equal in length to the line in plan, and whose perpendicu- 
lar is equal to the vertical hight of the object. 

Diagram of Triangles. 

In any convenient position we may draw lines as 
shown at a b, and b c, Fig. 9, which are at right angles 




Fig. 9. Diagram of tri- 
angles from which true 
lengths are obtained. 

to each other. Set off from b upon line a b, a distance 
equal to the vertical hight of the object as at d. Upon 
referring to the plan Fig. 8, we find that two of the lines 
radiating from the vertex of each angle of the square 
are of equal lengths, and since the upper extremities of 
these lines are all at the same distance from the base, it 
follows that the true lengths of lines of which these arc 
a plan, are equal in each group. 

Upon measuring lines as k d, and i d of the plan, they 
are found to be of equal lengths, therefore in this ex- 



10 



TRIANGULATION 



ample there are but two lengths to be determined, unless 
for convenience, we assume two additional lines as k n, 
and i o, thereby enabling us to designate one quarter of 
the object in plan, as k in i o d n, which may be duplicated 
for the other three equal parts. Having previously lo- 
cated point d, Fig. 9, we may set off from point b upon 
line b c, distances equal to lengths of lines k n, k d, and 
■in d, as in points p q and r. Upon drawing lines as 
shown at Fig. 9, we have in the length of line d p, the 
true length of a line of which k n is a plan, in d r the true 




Fig. 10. Pattern for one- 
quarter of the object. 

length of a line of which k d is a plan, and in d q the true 
length of a line of which m d is a plan. 

We are now in a position to develop a pattern, one 
quarter of which is shown at Fig. 10, since the plan Fig. 
8, and the diagram of triangles Fig. 9, supply all neces- 
sary measurements for that purpose. 

The Pattern. 

In any convenient position draw a line whose length 
is equal to length of line d q Fig. 9, as shown at m d 
Fig. 10. 

It may be remarked that this line is in reality the line 
shown in perspective at b d, Fig. 6, and that there are 



ELEMENTARY PRINCIPLES 11 

two additional lines radiating from point d, as shown in 
perspective by d a, and (/ c. The distances between the 
extremities of those lines at a and c, Fig. 6, are found 
in lengths of lines m k, and m i, Fig. 8. With compasses 
set to a span equal to the length of line d r, Fig. 9, and 
with point d, Fig. 10, as center, describe arcs as shown 
at k and i. With compasses set to a span equal to length 
of lines k m or m i. Fig. 8, and with point m, Fig. 10, as 
center, describe arcs as also shown at k and /, thereby 
locating those points in their correct relative positions. 

Presuming that only one quarter of the pattern is to 
be developed as shown, and that the seam is required to 
be upon a line as shown at k n or i o of Fig. 8, there is 
an additional line radiating from points k and i, as 
shown in plan at Fig. 8. The true lengths of these lines 
have been found in (/ p. Fig. 9, therefore the compasses 
may be set to a span equal to the length of that line, and 
with points k and i, Fig. 10, as centers, the small arcs 
may be drawn as shown at // and o. Since the plan, Fig. 
8, supplies the true length of one side of the triangles 
k n d, and i o d, the compasses may be set to a span equal 
to the length of line n d or d o, Fig. 8, and with point d, 
Fig. 10 as center, describe the small arcs cutting the first 
at points ;/ and o, when lines may be drawn as shown, 
which completes the pattern for one quarter of the 
object. 

This is all that is necessary, since it may be duplicated 
for the remaining three equal parts, or the lengths of 
lines as shown may be used in rotation to develop the 
whole pattern. 



CHAPTER II. 

A Simple Transitional Fitting from Square to 

Round. 

When the sheet metal worker is called upon to secure 
the pattern for a fitting as illustrated at Fig. 11, i. e., 
from square to round, with the center of the top directly 
above the center of the base, he may employ the principles 
explained in Chapter I. 




Fig. 11. Photographic View of a Fitting of Common 
Occurrence in the Sheet Metal Shop. 

In all examples of pattern development, where some 
portion of the object for which a pattern is required is 
represented by a circle, said circle must be divided into 
parts. The points of division forming these parts, are 
in reality, the vertices of angles forming a polygon of 

12 



FROM SQUARE TO ROUND 



13 



as many sides as the parts into which the circle has been 
divided. The vertices of angles at points e f g h, etc., of 
Fig-. 8, Chapter I, may be looked upon as points of 
division in a circle whose diameter is equal to the major 
diameter of the polygon. However, since eight parts are 
too few to divide a circle into, we would have divided 
that circle into a greater number. The author has found 
sixteen to be quite effective, although a still greater 
number will more closely approximate the circle. 

A Plan Drawn to Given Dimensions. 
Presuming that it is required to secure the pattern for 
a fitting as illustrated at Fig. 11, to given dimensions, 
the square A B C D, Fig. 12, is drawn to the size of the 




12. The Plan of the Fitting 
Illustrated in Fig. 11. 

base, and its center located as at w. With point w as 
center, a circle is drawn equal to the required diameter 
of the top, as shown at 1 2 3 4, etc. Since this is an 
example of Triangulation, the surface of the object is 
presumed to be divided into triangles. 

To secure the plans of said triangles, the circle is 



14 TRIANGULATIOX 

divided into parts as shown. This is accomplished by 
first dividing the circle into four parts by lines parallel 
to the sides of the square, as shown by the dotted lines, 
1 9 and 5 13. Each quarter of the circle is then divided 
into four equal parts. This forms the points of division 
of the circle into four groups, as 1 5, 5 9, 9 13, and 13 1. 
Lines drawn from the points of division in each group 
to the vertex of the adjacent angle as shown, will com- 
plete a plan of the fitting, together with the plans of 
triangles, which in reality, form the required pattern. 

Size and Form of Triangles. 

Since, as has been previously explained, the triangles 
whose plans are shown in the diagram, Fig. 12, form 
the pattern, we must determine their exact size. This 
is accomplished by securing the lengths of lines which 
form said triangles. As will be noted, the plan supplies 
the length of one side of each, i. e., there are four whose 
bases form the square, and the length of one side of 
each remaining triangle is found in the distances the 
points of division of the circle are from each other. 
Therefore we have simply to determine the true lengths 
of lines forming the remaining two sides of each tri- 
angle shown. This is still further simplified in exam- 
ples where the form is symmetrical as in this case, inas- 
much as corresponding lines in each quarter of the plan 
are of the same length. 

Upon examination, we find that lines 1 D and 5 D 
are equal in length, also 2 D and 4 D. Therefore there 
are but three lengths to be determined, as 1 D, 2 D and 
3 D, unless we introduce two additional lines as 1 x and 
5 y, to locate one-quarter of the plan, or the seam, thus 
making one additional length to be determined, as 1 x. 

The student should have little difficulty in comprehend- 



FROM SQUARE TO ROUND 



15 



ing the lines shown in plan, i. e., 1 x, 1 D, 2 D and 3 D, 
inasmuch as they are presumed to be upon the surface of 
the object. The upper extremities of said lines are at 
points 1 2 and 3 of the top, and their lower extremities 
are in points x and D at the base; therefore in reality 
these lines are inclined to the base. If we dropped lines 
from points 1 2 and 3, perpendicular to the plane of the 
base, their intersections with that plane will locate points 
whose distances from point D have previously been de- 
termined in the plan. The student may refresh his mem- 
ory upon this by referring to Fig. 6, Chapter I, where 
perpendicular lines are shown in perspective as a e and 
b f, which are in reality equal in hight to the vertical 
hight of the object. Thus it will be noted that such 




Fig. 13. Diagram of Triangles 
from which True Lengths are 
Secured. 

lines supply the necessary lengths of perpendiculars for 
all triangles employed in securing the true lengths of 
those lines which are in reality inclined to the planes of 
the top and base of the object. 

The distances these lines (i. e., perpendicular lines 



16 TRIANGULATION 

as shown at Fig. 6) are from point D at their intersec- 
tion with the plane of the base have previously been de- 
termined in the plan, Fig. 12; we therefore use the 
lengths of lines 1 x, 1 D,2 D and 3 D as the bases of said 
triangles. 

On the Construction of Triangles. 

To construct the necessary triangles, draw the indefi- 
nite right lines at right angles to each other, as E F and 
F G, Fig. 13, which intersect at point F. Set off upon the 
vertical line from F, a distance equal to the vertical 
hight of the object, as shown at E. Set off also from F 
upon the horizontal line, distances equal to the lengths 
of lines 1 x, 1 D, 2 D and 3 D, found in Fig. 12, as 
shown by points x, 3, 2, and 1, Fig. 13. From these 
points lines are drawn to E as shown, thus securing the 
lengths of all lines necessary to develop the pattern, since 
the plan supplies the remaining lengths. 

The Pattern. 

To develop one-quarter of the pattern as shown at 
Fig. 14, draw the line 3 D, making it of a length as 
found at 3 E, Fig. 13. With compasses set to a span 
equal to distances between points 3 and 2, or 3 and 4, of 
Fig. 12, place one point at 3 of the pattern, and describe 
small arcs as shown at 2 and 4. With compasses set to 
a span equal to length of line 2 E, Fig. 13, place one 
point at D, and describe small arcs as also shown at 2 
and 4, thus locating these points in their correct relative 
positions. 

From points 2 and 4 as centers, small arcs are drawn 
with a radius equal to the distance between points 2 and 
1, or 4 and 5. With point D as center, and with a radius 
equal to the length of line 1 E, Fig. 13, describe arcs as 



FROM SQUARE TO ROUND 



17 



also shown at 1 and 5, thus securing the correct relative 
positions of those points. 

Upon referring to the plan, we note that there are two 
additional triangular surfaces to be added to complete 
the pattern for one quarter of the object as shown; 
these are the triangles bounded by lines 1 x, x D and D 1, 
also 5 D, D y, and y 5. Since the plan supplies the true 
lengths of one side of these triangular surfaces, we set 
our compasses to a span equal to length of line x D or 
y D of plan, place one point at D of the pattern, and 
describe arcs as shown at x and y. With points 1 and 5 




Fig. 14. Pattern for One Quarter of the 
Fitting as Illustrated at Fig. 11. 

as centers, and the length of line E x, Fig. 13, as radius, 
place one point at points 1 and 5, and describe arcs as 
also shown at x and y, thereby locating these points. 
Lines are now drawn as shown, thus locating the 
boundaries of triangles, which, having been placed in 
their correct relative positions, supply the required pat- 
tern for one quarter of the object, which may be dupli- 
cated for the remaining three equal parts. 



lg TRIANGULATION 

The Necessity of a Clear Conception of the Prin- 
ciples Involved. 

If the student fails to secure a clear conception of the 
principles as here set forth, he is earnestly advised to 
review the work, since the author has made an honest 
endeavor to explain the most elementary principles in- 
volved. To the one who aspires to become proficient in 
this branch of pattern development, a knowledge of 
those principles is of the utmost importance. The prac- 
tice has been too often to work from copy rather than to 
make a study of the principles involved. This is hardly 
more calculated to make a pattern cutter, than so much 
time spent in copying music would be to make a mu- 
sician. It is precisely the same with the subject in ques- 
tion; the various lines employed in pattern development 
can be copied by almost any novice, but no amount of 
copying will enable him to understand them. On the 
other hand, if one becomes thoroughly conversant with 
the principles involved, and secures a clear understand- 
ing of the meaning of every line drawn, the solving of 
complicated problems is but a simple operation. 



CHAPTER III. 
The Oblique Cone. 

Attention will now be directed to the Oblique Cone, 
and the method of securing the pattern for same. This 
is a simple example containing principles which may be 
employed in securing the patterns for a variety of forms. 

In the compilation of this work, the author has as- 
sumed no previous knowledge of orthographic projection 
on the part of the student. As the examples demand 
some knowledge of this, an explanation of the principles 
involved will be entered into as the work progresses. 

The Plan. 

A plan is denned as being a drawing of anything, 
showing the parts in their proportion and relation. The 
surface upon which a plan is drawn is a horizontal one, 
and in this work we shall presume that the object to be 
represented is directly above it. 

In securing the location of points which it becomes 
necessary to represent in plan, vertical lines are presumed 
to be dropped from said points, and their intersections 
with the surface upon which the plan is drawn, are the 
plans of those points. 

Fig. 15 illustrates an oblique cone with a number of 
right lines upon its surface, and its plan. Fig. 16 is a 
geometrical representation of a similar cone, and is 
looked upon as a plan. 

The purpose of Fig. IS is to convey to the student in 
a pictorial way, an understanding of the relation the plan 
bears to the object itself. Here, as will be noted, an 

19 



20 



TRIANGULATION 



oblique cone with a number of right lines upon its sur- 
face is suspended directly above the horizontal surface 
A B C D. The method of securing a plan of said cone 
should be apparent from lines shown. As will be noted, 
the foot of the perpendicular line E E' is a plan of the 




Fig. 15. An Oblique Cone Having a Number of Right Lines 
Upon Its Surface, and Its Plan, 

vertex E, and the foot of each perpendicular let fall from 
the numbered points of the base is the plan of its respec- 
tive point. Since lines shown upon the surface of the 
cone are right lines, and converge to the vertex E, lines 



THE OBLIQUE CONE 21 

may be drawn from numbered points of the plan to E' , 
to secure plans of those lines. 

Dividing the Surface of the Oblique Cone. 

When called upon for a pattern for an oblique cone, 
certain data must be at hand in the form of a specifica- 
tion, i. e., the diameter of the base, the vertical hight 
of the vertex above the plane of the base, and the distance 
the vertex is removed from a point directly above the 
center of the base. Presuming these to be known, and 
to be as shown at Fig. 16, where the circle is drawn equal 
in diameter to the diameter of the required cone, a line is 




Fig. 16. A Plan of an Oblique Cone. 

drawn through the center of said circle as 1 9. Upon this 
line a point as A is located, which is at a distance from 
the center of circle equal to the distance the vertex of 
the cone is removed from directly above the center of 
the base. 

Since the line 1 9 divides the diagram into two equal 
parts although opposite, we shall only consider one part, 
as it may be duplicated for the other. One-half of the 
circle is divided into a number of equal parts as shown, 
and lines drawn from these points of division to point A. 



22 



TRIANGULATION 



In this manner the plan is secured of not only the cone, 
but of a number of right lines upon its surface, which 
divide said surface into triangles. 

The work of securing the pattern is a matter of de- 
termining the dimensions of these triangles, and placing 
them upon any surface, a portion of which will then 
constitute the pattern. ' Since the distance between points 
of the circle is the true length of one side of each tri- 
angle, the remaining measurements are secured by de- 




Fig. 17. Diagram of Triangles. 

termining the true lengths of lines which connect points 
of the base and the vertex. As said lines intersect at 
point Ej Fig. 15, the true length of each will be found 
in the hypothenuse of a right angled triangle, whose 
perpendicular is equal to the vertical hight of the re- 
quired cone. 

Constructing Necessary Triangles. 

The method of constructing these triangles is clearly 
shown at Fig. 17, where indefinite right lines A B and 
B C have been drawn at right angles to each other, and 



THE OBLIQUE CONE 



23 



intersecting at point B. The vertical hight of the re- 
quired cone is set off from B upon the line A B, as at A. 
Since each line shown in plan, Fig. 16, i. e., A 1, A 2, 
etc., supplies the true length of the base of a triangle, 
whose hypothenuse is the line in its true length, we may 
set off from B along line B C, distances found from A, 
Fig. 16, to points 1, 2, 3, etc., as shown in similarly num- 
bered points of Fig. 17. The distances found from point 
A, Fig. 17, to the numbered points upon line B C, are 
the true lengths of similarly designated lines whose plans 
are shown in Fig. 16. Thus all necessary data is at hand 
to enable us to complete the pattern. 




Fig. 18. Semi-pattern for an Oblique 
Cone. 

The vertices of all triangles of which the surface of 
the cone is composed, are at the apex of said cone, there- 
fore the lines forming two sides of each of those triangles 
will radiate from point A of the plan, or pattern, as 
shown. 



24 TRIANGULATION 

To Secure the Pattern. 

To develop the pattern as shown at Fig. 18, draw the 
line A 1 in any convenient position, making its length 
equal to the length of line A 1, Fig. 17. Since line A 2 
also radiates from point A of the pattern, the compasses 
may be set to a span equal to the length of line A 2, Fig. 
17, and with point A of the pattern as center, the small 
arc is drawn as shown at 2, Fig. 18. Then will one ex- 
tremity of line A 2 lie in said arc, and at a distance from 
point 1 equal to the distance between points 1 and 2 of 
the plan, Fig. 16. Therefore, if the second arc is drawn 
as shown, with point 1 as center, and the distance be- 
tween points 1 and 2 of the plan, Fig. 16, as radius, the 
exact location of said line is established. The true size, 
form, and position of the remaining triangles of which 
the surface of the cone is composed, may be determined 
in a similar manner. For example, point A of the pat- 
tern, Fig. 18, is a constant center from which small arcs 
are drawn whose radii are equal to lengths of lines 
A 3, A 4, A 5, etc., Fig. 17. Successive numbered points 
of the base are used as centers, with the distance simi- 
larly numbered points are from each other as shown in 
plan, Fig. 16, as radii, to locate those points as shown, 
Fig. 18. 

Fig. 18 shows the pattern for one-half the cone which 
may be duplicated for the remaining equal portion. Had 
it become desirable to secure the pattern in one piece, 
points and lines as here shown could have been duplicated 
upon the opposite side of line A 1. 



CHAPTER IV. 

A Transitional Fitting from Rectangular to 
Round Which Makes An Offset. 

Fig. ,11, Chapter II, illustrated a fitting making a 
transition from square to round, with the center of the 
top directly above the center of the base. The sheet metal 
worker is frequently called upon for a fitting making a 
similar transition, but whose top is not in the same rela- 
tive position, i. e., it may be what is ordinarily termed 
straight on one side, or it may be required to offset. In 
examples of this description, there is little or no varia- 
tion in the methods to be employed. The same principles 
are involved, providing the ends are parallel. 

The Specification. 

From the specification a conception of the object is 
secured. This is purely a mental process ; clear concep- 
tions may be formed in the dark, or by one blindfolded. 
Some difficulty may be experienced by the novice in 
forming clear conceptions of the objects from their speci- 
fications, although the power of doing so is essential in 
pattern development, since before an object can be rep- 
resented, it must be known what that object is. This 
power may be cultivated and increased by practice. The 
conception is formed from the specification, by knowing 
the size and form of the base, the size and form of the 
top, and the distance the plane of the top is above the 
plane of the base, also the position the top is required to 
occupy as regards the base. 

25 



26 



TRIANGULATION 



The student is advised to look upon Fig. 19 as the 
specification for a fitting, the pattern of which is re- 
quired. As indicated, the base is to be rectangular and 
12 x 16 inches in size, the fitting to have a round top 10 
inches in diameter. The vertical hight of the object is 
to be 16 inches, i. e., the perpendicular hight between 
the planes of the top and base, is 16 inches. The center 
of the top is located directly above a line which divides 
the rectangle longitudinally into two equal parts as 
shown, but 7 inches from its central point. 

To Draw the Plan. 

The student may picture the above conditions in his 
mind, and we will proceed to draw the necessary plan. 
To represent the base in plan, draw the rectangle A B 




Fig. 19. A Pictorial Specification for a Fitting. 

D C as shown at Fig. 20, with lengths of sides of 12 and 
16 inches. Fig. 20 includes a scale to which the dia- 
grams in this problem have been drawn (i. e., Figs. 20, 
21 and 22), and if the student desires, he may take meas- 
urements as there shown, and apply to those diagrams as 
we proceed with the explanation. 



RECTANGULAR TO ROUND 



27 



A line is drawn through the center of the rectangle, 
and parallel to its longest side as x y. The center of the 
rectangle is, of course, the center of the line x y, as shown 
at z. Locate a point upon line x y, 7 inches from point z 
as w, then will point w be a plan of the center of the top, 
or a point to be used as a center about which a 10 inch 
circle is to be drawn. This circle is a plan of that end, 
and completes a plan of the fitting. 

Plans of Triangles. 

To secure the plans of triangles, which when combined 
constitute the surface of the fitting, or its pattern, the 
circle is first divided into four equal parts by lines parallel 




Scale 



' i T l T-T l T |! [' |, l ,| T l T l T l 'l' |, r i T l T l T' l T l T l 'r j - J l 1|, l J| ' 

3 6 9 IF. 15 18 



Fig. 20. The Plan of a Fitting, together 

with the Scale to zvhich it has been 

Drawn. 

to the sides of the rectangle, as shown in points 15 9 
and 13. Divide each quarter of the circle into the same 
number of equal parts as also shown. Draw lines from 
all points thus secured in each quarter of the circle, to 
the vertex of its adjacent angle of the rectangle, as 
shown at 1 C, 2 C \ 3 C, 4 C, 5 C, 5 D, 6 D, etc This 
completes the plans of the above spoken of triangles, 
which constitute the whole surface of the fitting. How- 



28 TRIANGULATION 

ever, since the line x y divides the plan into two equal 
parts, the covering for one part will, when reversed, or 
formed in the opposite direction, supply a covering for 
the remaining part, therefore it is only necessary to con- 
sider one half of the plan. Several of these triangles are 
shown in perspective in Fig. 19, and by the aid of this 
figure the student should have little difficulty in compre- 
hending the positions of said triangles upon the surface 
of the fitting. 

The Value of Perspective. 

The object of the author in presenting Fig. 19 has not 
only been to present a specification in a pictorial way, but 
to illustrate triangles whose plans are shown in Fig. 20. 
With the plan of each triangle before us, which when 
combined with all the others, will constitute the surface 
of the fitting, or its pattern, the next step is to determine 
the true form and size of each, and place them upon the 
surface, a portion of which will constitute the pattern. 

Another logical deduction which may be applied to 
examples in pattern development, is to consider each line 
separately, and determine their true lengths and relative 
positions. 

To Determine the Size and Form of Each Com- 
ponent Triangle. 

The size and form of a triangle can be determined if 
the lengths of the three sides of which it is composed 
are known. Thus the question rests upon our ability to 
determine the true lengths of lines shown in plan, and 
the relative position these lines should occupy when 
placed upon the plane of development. Here, as in fore- 
going examples, the true lengths of a considerable num- 
ber of the lines are shown in plan, i. e., those lines which 



RECTANGULAR TO ROUND 



29 



form the outline of the top and base. However, those 
lines which connect points of the base with points of the 
top are not shown in their true lengths and must be de- 
termined by the use of the right angled triangle, as 
shown at Fig. 21. 



To Determine the True Lengths of Lines. 

The- true lengths of lines as shown at Fig. 21 are de- 
termined by drawing the lines E F and F G at right 
angles to each other intersecting at point F. From F 
upon line E F, set off a distance equal to the vertical 
hight of the fitting (16 inches). From F upon the line 
F G, set off distances equal to lengths of lines in plan, 




Fig. 21. A Diagram of Triangles 
from which True Lengths have been 
Secured. 

Fig. 20, as 1 C, 2 C , 3 C, etc., and from said points draw 
lines to point E. Then will each line represent the true 
length of its respective line in plan. Upon completing the 
diagram of triangles as shown at Fig. 21, we may proceed 
to develop the pattern as follows. 

To Develop the Pattern. 
The true length of line 1 x shown in plan, Fig. 20, is 
found in line 1 x, Fig. 21, therefore we may place that 



30 



TRIANGULATION 



line in its true length in any convenient position as at 1 x 
of the pattern, Fig. 22. The line x C of the plan is there 
shown in its true length, therefore we may set the com- 
passes to a span equal to the length of line x C, Fig. 20, 
and placing one point at x of pattern, Fig. 22, describe 
the small arc as at C. With the length of line C 1, Fig. 
21, as radius, and with point 1 of pattern as center, de- 




Fis. 22. 



The Semi-pattern for a Fitting 
as Illustrated at Fig. 19. 



scribe the second small arc at C of the pattern, Fig. 22, 
thereby locating that point, or line 1 C, in its correct 
relative position. 

Upon referring to the plan, Fig. 20, we note that there 
are four additional lines radiating from point C, and that 
the distances between the upper extremities of those lines 
are equal to the distances between points of division of 
the circle. Therefore we use the lengths of those lines in 
rotation, i. e., C 2, C 3, C 4, and C 5, Fig. 21, as radii, 



RECTANGULAR TO ROUND 31 

with point C, Fig*. 22, as center, to describe small arcs as 
shown at 2, 3, 4 and 5 of pattern, Fig. 22, then with dis- 
tances between numbered points of the circle as shown in 
plan, Fig. 20, as radii, and using the numbered points 
of the pattern in rotation as centers, we describe small 
arcs as also shown at 2, 3, 4 and 5 of the pattern, Fig. 22. 

We note that point D upon the surface of the object 
is at a' distance from point C equal to the length of line 
C D of the plan, Fig. 20. Therefore we may set our 
compasses to a span equal to the length of line C D of 
the plan, Fig. 20, and placing one point at point C of the 
pattern, Fig. 22, describe the small arc as at D. Point D 
of the pattern must then lie in some point of this arc. 
The true distance from point 5 to D upon the surface of 
the object as shown in perspective at Fig. 19, and in plan, 
Fig. 20, is the length of line D 5 in the diagram of tri- 
angles, Fig. 21. Thus we may use the length of this line, 
i. e., D 5 of the diagram of triangles, as radius, and point 
5 of the pattern, Fig. 22, as center, to describe the second 
small arc as shown at D, thereby locating that point, or 
line D 5, in its correct relative position. 

The plan, Fig. 20, clearly shows four additional lines 
radiating from point D. The upper extremities of these 
lines are at a distance from each other equal to the dis- 
tance between points of the circle. Therefore we may 
use the true lengths of these lines in rotation, which are 
found in the diagram of triangles, Fig. 21, i. e., D 6, D 7 , 
D 8, and D 9 as radii, and with point D of pattern as 
center, to describe small arcs as shown at 6, 7 , 8 and 9, 
and the distances between similar numbered points of 
the circle, Fig. 20, as radii to be used successively with 
points 5, 6, 7 and 8 of the pattern, Fig. 22, as centers, to 
locate these points in their correct relative positions, as 
shown. 



32 TRIANGULATION 

The plan, Fig. 20, or the pictorial view of the object, 
Fig. 19, shows a triangular surface as 9 D y, which may 
now be added. Since the true length of line D y is shown 
in plan, we may use that length as radius, with point D 
of the pattern, Fig. 22, as center, to describe the small 
arc as at y, then will point y of the pattern lie in some 
point of this arc. 

We find that the true distance from point 9 to y, is the 
length of line y 9 of the diagram of triangles, Fig. 21. 
Therefore we use that length as radius, with point 9 of 
the pattern, Fig. 22, as center, to describe the second 
small arc as shown at y of the pattern, thus locating that 
point in its correct relative position, which completes the 
pattern for one-half the fitting. 

The Order of Numbering May Be Reversed. 

It may be here explained that while the author has 
designated one end of the longest line presumed to be 
upon the surface of the object as 1, he could as consistent- 
ly have reversed the order of numbering, thereby placing 
1 in the position now occupied by 9, i. e., it makes no 
material difference which part of the pattern is first de- 
veloped. 

It may be further explained that each part of the plan, 
Fig. 20, i. e., those parts included between points 1 C 5, 
or 5 D 9, may be compared to an inverted oblique cone, 
and that the patterns for those portions are developed in 
the same general manner as for the oblique cone. 



CHAPTER V. 
A Twisted Transitional Fitting. 

Attention is here directed to a form as illustrated at 
Fig. 23, which is an excellent example for practise, since 
it may be represented by a comparatively few lines. 

It may be here remarked that it requires far more 
study to conceive the forms whose patterns may be de- 




Fig. 23. 



The View of a Fitting whose Form is Somewhat 
Unusual. 



veloped by triangulation, than to secure an understand- 
ing of the few principles involved after the form has 
been conceived. Therefore the best advice to be given 
to those who desire to secure a clear understanding of 
this branch of pattern cutting is to give the unusual form 

33 



34 



TRIANGULATION 



the same careful attention that they may devote to the 
more common ones. The fitting, as illustrated at Fig. 
23, is in reality the connection between two rectangular 
pipes whose forms of cross-section are not identical yet 
of approximately the same area. It will be noted that 
the sides of said pipes are not parallel, i. e., the fitting 
performs a twist. 

Fig. 24 is a perspective view of a fitting of this descrip- 
tion and its plan. This has been introduced for the pur- 




Fig. 24. The Fitting and Its Plan, Shown 
in Perspective. 

pose of enabling the student to secure an understanding 
of the relation the plan bears to the object itself, since it 
may be somewhat difficult to form a conception of the 
object from its plan, Fig. 25. 

Principles Which Govern the Work of Drawing 

a Plan. 
It should be remembered that Fig. 25 is a geometrical 
representation, and the one which must be employed to 



A TWISTED FITTING 



35 



secure the pattern, while Fig. 24 is a perspective view of 
the object which is presumed to be suspended directly 
above the horizontal surface E F G H. As will be noted, 
vertical lines have been dropped from points of the object 
to intersect this surface, thereby illustrating" the princi- 
ples which govern the work of drawing a plan as shown 
at Fig. 25. 

From the above, and by the aid of Fig. 24, the student 
will readily understand that the rectangles, 12 3 4 and 




Figs. 25 and 26. 25, A Plan of Fitting; 
26, Diagram of Triangles. 

A B C D, are the cross-sections of pipes to be connected. 
These have been placed in the same relative positions that 
said pipes or collars would occupy if the object was 
viewed from above, and with the point of sight moving 
in such a manner as to bring every point viewed in a 
line perpendicular to the horizontal surface upon which 
the plan is supposed to be drawn. 



36 TRIANGULATION 

It may be remarked that only the irregular portion is 
being considered, since the collars at each end are here 
looked upon as separate and independent parts, whose 
patterns do not involve triangulation. 



The Surface of the Fitting. 

We note that the surface of the fitting is made up of 
eight triangles. These triangles, when combined and 
placed in their correct relative positions upon a flat sur- 
face, will constitute the pattern, therefore their true 
form and dimensions must be determined. This, as will 
be noted, makes the plan an important factor in the solu- 
tion of the problem. 

The plan of each triangle of which the surface of the 
object is composed, is secured by drawing lines from 
each angle of the rectangle A B C D, to two adjacent 
angles of the rectangle 12 3 4, or conversely, from each 
angle of the rectangle 1 2 3 4 to two adjacent angles of 
the rectangle A B C D. Thus a plan of the fitting is com- 
pleted, as shown at Fig. 25. 

It will be noted that points 12 3 4 are at the base, and 
points A B C D are at the top of the object, therefore 
lines as A 1, A 3, B 2, etc., are the plans of lines which 
connect points of the base to points of the top, and are 
oblique to the planes within which the top and base are 
situated. However, since we know the vertical hight of 
the object, we know the vertical distance between the ex- 
tremities of those lines. This distance is the length of 
one side of all triangles which it becomes necessary to 
construct to secure the true length of lines connecting 
points of the base with points of the top, which are the 
boundaries of triangles upon the surface of the fitting, 



A TWISTED FITTING 



37 



Size and Form of Triangles. 

If the true form, size, and relative positions of said 
triangles can be determined, they may be placed upon a 
flat surface in those sizes and positions to complete a 
pattern. The dimensions of said triangles will be de- 
termined in the same manner as has been previously ex- 
plained, i. e., the lengths of those lines which form sides 
of the above spoken of triangles, and connecting points 
of the base to points of the top, are found by the use of 
the right angled triangle, as shown at Fig. 26. Here 
the lines E F and F G are drawn at right angles to each 
other, intersecting at the point F. The vertical hight 




Fig. 27. Pattern. 



of the object is set off from point F upon the line F E, 
as at E. 

The lengths of lines shown in plan as 1 A, 3 A, 3 C, 
4 D, etc., are set off upon line F G from point F, as 
shown. Lines are drawn from said points to point E to 



38 TRIANGULATION 

supply the true lengths of lines shown in plan, or those 
lines which connect points of the base to points of the top. 
The lengths of those sides of the triangles which form 
the base or top of the fitting, are found in their true 
lengths in plan, since the true forms of pipes to be con- 
nected are there shown. 

The Pattern. 

With these lengths determined, the pattern is secured 
as shown at Fig. 27, where, to economize space, it is 
shown in two parts, a portion of one part being repre- 
sented by dotted lines. The student who has given at- 
tention to the above will note that points ABC and D 
are at angles of the top, and shown in each view, i. e., if 
the pattern was wrapped about the object, these points 
would occupy positions as represented in the several 
views. 

On Dividing Diagrams which Represent the Ends 
of the Object. 

When each end of the object for which a pattern is 
required can be represented by rectilinear diagrams, 
there is no necessity of dividing said diagrams into 
parts, since the vertices of their angles are used as points 
of division. However, when one or both of its ends must 
be represented by a curvilinear diagram, said diagram 
must be divided into parts. 

Here the student should recognize the fact that these 
points of division are in reality, points upon the end of 
the object, to which lines are presumed to be drawn from 
points upon the opposite end. Since these lines are con- 
sidered as straight lines, they should be so located as to 
allow them to be straight when placed upon the object. 
If this is not accomplished some error must exist in the 



A TWISTED FITTING 



39 



pattern, and if there is considerable variation in these 
lines, i. e., if they are presumed to be straight and are 
so located as to cause them to be considerably curved, 
there must be some distortion in the fitting when made 
from the pattern. Tt is quite possible to so locate these 




Plans of Fittings. 



lines as to preclude the pattern being formed into its 
required shape without a stretching or drawing of the 
material. Each individual case requires some attention 
to this, as it is a difficult matter to apply a fixed rule 
to all. 



40 TRIANGULATION 

Fig. 28 contains diagrams which may be looked upon 
as plans of fittings. No. 1 is a fitting making a transi- 
tion from rectangular to round, the round end being so 
placed as to require the lengths to be secured of prac- 
tically all lines presumed to be upon its surface and 
shown in plan, i. e., the plan cannot be divided into 
equal parts. No. 2 is a plan of a fitting from rectangular 
to elliptical; the major axis of the ellipse has here been 
placed directly above the diagonal of the rectangle, there- 
fore this diagram could be divided into two equal parts. 
However, as here shown, the better course would be to 
determine the true lengths of all lines. No. 3 is the plan 
of a fitting whose form and size of its ends are the same 
as those shown at No. 2, but not in the same relative 
positions. Here, as with No. 2, the better course to 
pursue when developing the pattern, is to determine the 
true lengths of all lines shown, or the true form and size 
of all triangles of which its surface is composed. 

To locate triangles presumed to be upon the surface 
of fittings whose plans are at Nos. 1, 2 and 3, the curvi- 
linear figures are divided into parts, and said points of 
division should be so located as to allow right lines to 
be drawn upon the object from the corners of one end to 
these points at the other. This is accomplished by di- 
viding the diagrams as shown. The points A B C D in 
each are the important ones, and having located these 
in satisfactory positions, the intermediate points as a b 
c d, etc., may be located at pleasure, i. e., each part of 
the curve contained between points ABC and D may be 
divided into any convenient number. 

After having located lines whose approximate posi- 
tions are shown at Fig. 28, the process of securing the 
pattern is substantially the same as has been explained 
in Chapter IV. 



CHAPTER VI. 
The Pattern for the Frustum of an Oblique Cone. 




Fig. 29. The Frustum of an Oblique Cone. 

As has been explained, the surface of the object for 
which a pattern is required, may be presumed to be 
divided into triangles. In foregoing chapters, forms 
have been selected whose rectilinear elements divide said 
surfaces into triangles. Attention will now be directed 
to forms whose rectilinear elements of their surfaces do 
not divide said surfaces into triangles, therefore addi- 
tional lines must be introduced. 

41 



42 TRIANGULATION 

Elements of a Surface. 

It may be here explained, that lines drawn, or pre- 
sumed to be drawn upon the surface of the cone or 
cylinder, are termed elements of that surface, and if 
drawn in positions which admit of their being right 
lines, they are known as rectilinear elements. 

Thus we note upon referring to Fig. 12, Chapter 2, 
that lines D 1, D 2, D 3, etc., are plans of rectilinear 




Scale 

T I T I T I T | I I I T'T I T I T |I I I I |I I I I I T I T I T I T I T I T | 1 
3 6 9 IF 16 18 



Fig. 30. The Plan of a Fitting 
Illustrated in Fig. 29. 

elements presumed to be upon the surface of the object 
represented. Similarly, upon referring to Fig. 16, 
Chapter III, lines as 1 A, 2 A, 3 A, etc., are plans of 
rectilinear elements of the conical surface. 

Pattern for the Frustum of an Oblique Cone. 

Presuming the pattern is required for a fitting as illus- 
trated at Fig. 29, the specification must supply the di- 
ameters of the top and base, together with its hight and 
the relative positions of its ends. In this example, it 
has been presumed that one side is perpendicular to the 



FRUSTUM OF AN OBLIQUE CONE 



43 



plane of its ends, or, what is sometimes termed, straight 
on one side. This form is the frustum of an oblique cone. 
The most simple and efficient diagram which will repre- 
sent the object is a plan as shown at Fig. 30. This as- 
sumes the object to occupy a position as shown at Fig. 31. 

scenographic and orthographic projection 
Compared. 

It may be well to here explain that Fig. 31 is a pictorial 
view of the object, and its plan. This is a scenographic 
representation and of no particular value beyond its use 




Fig. 31. Pictorial View of the 
Fitting and Its Plan. 

to convey to the reader an understanding of the position 
the object occupies as regards its plan. 

In continuation of the above, it may be stated that in 
representing objects according to the principles laid 
down for perspective, the eye is imagined to be stationed 
in one particular place, called the point of sight, from 



44 TRIANGULATION 

which all the visible parts of the figure are supposed to 
be seen. In orthographic projection, with which we are 
chiefly concerned, the case is very different, inasmuch as 
the eye is supposed to be in a direct line with every part 
viewed, or, in other words, to move over the object in 
such a manner as to be directly opposite to every part 
represented. The visual rays are therefore parallel, 
whereas in perspective they converge to a point. 

A Geometrical Representation, or a Plan. 

Fig. 30 is a geometrical representation or a plan of 
the object, which has been drawn to the scale appended, 
presuming the dimensions to be as follows: Diameter 
of base, 20 inches; diameter of top, 14 inches; hight, 
2iy 2 inches, with one side perpendicular to the plane of 
its ends. 

The circles have been placed in the same relative posi- 
tion as the ends of the object would appear if viewed 
from above as in orthographic projection. A line as 1 9, 
drawn through the center of each circle, divides the 
plan into two equal parts, therefore it only becomes 
necessary to consider one part. As will be noted, one 
half of each circle has been divided into an equal number 
of equal parts, as 1, 2, 3, etc. Lines drawn between 
points of the same number in each circle as shown, sup- 
ply plans of lines which are presumed to be upon the 
surface of the object. The above spoken of lines are 
clearly shown in perspective at Fig. 31. 

As has been previously explained, the development of 
the pattern rests upon our ability to determine the 
lengths of these lines and place them upon a flat surface 
in their correct relative positions. The plan supplies 
the distances these lines are from each other at their 
extremities in the points of division of the circle. 



FRUSTUM OF AN OBLIQUE CONE 



45 



The Right Angled Triangle. 

By the use of the right angled triangle, we may secure 
the lengths of these lines as has been explained in fore- 
going chapters, i. e., draw two indefinite right lines at 
right angles to each other, as A B and B C, Fig. 32, and 
set oft' from B upon the line A B, a distance of 21*4 
inches, as at A. Set off from B upon the line B C, dis- 




\ > ,\\\r— 2-3 
\ \y» — 7^, 

\ \ \ii F ' 



Fig. 32. Diagram of Triangles. 



tances equal to lengths of lines 11,22,33, etc., which 
are shown in plan Fig. 30, as shown upon line B C, Fig. 
32. Then will the distances from A to those points upon 
line B C, supply the true lengths of similarly numbered 
lines shown in plan at Fig. 30, or in perspective at 
Fig. 31. 

Additional Lines Must Be Assumed. 

Upon attempting to develop the pattern with the data 
now before us, we find that these lines cannot be placed 
in their correct relative positions upon a flat surface. 

To enable the reader to realize this, we may attempt 
to develop the pattern by drawing a line in any con- 



46 



TRIANGULATION 



venient position, as 1 1 of the pattern, Fig. 33, whose 
length is shown in the distance between points A a, Fig. 
32, which is known to be the true length of line 1 1 upon 
the surface of the object. With the extremities of this 
line as centers, draw arcs whose radii are equal to the 
distances between points 1 and 2 of the large and small 
circle of the plan, Fig. 30. Since the radii of these arcs 
are in reality the distances between the extremities of 




Fig. 33. Semi-pattern. 

lines 1 1 and 2 2 upon the surface of the object, the ex- 
tremities of line 2 2 must lie in points of said arcs. As 
these arcs may be conceived as being formed of a great 
number of points, no two of which are in the same 
position, although at the same distance from their cen- 
ters, which are the extremities of line 1 1, it is yet a 
difficult matter to accurately locate the extremities of 
line 2 2. 



FRUSTUM OF AN OBLIQUE CONE 47 

However points 1 and 1, Fig. 33, have been located in 
definite positions. Since points 2 and 2 must lie in arcs 
which have been drawn with points 1 and 1 as centers, if 
the true distance between points 1 at the base and 2 at the 
top of the object was known, we could then locate point 2 
upon the pattern. This applies as well to practically all 
designated points shown in plan. 

To determine those lengths we draw lines as 1 2, 2 3, 
3 4, etc., Fig. 30, thus securing the plans of lines con- 
necting those points. As has been frequently explained, 
the lengths of those lines will be secured by the use of 
the right angled triangle as follows: Draw lines D E 
and E F, Fig. 32, at right angles to each other. Set off 
from E upon the line E D, a distance equal to the vertical 
hight of the object as at D. Set off from E along the 
line E F, distances equal to the lengths of lines 12,2 3, 
3 4, etc., Fig. 30. Then will the distances from those 
points to point D represent the true lengths of those lines, 
i.e., 1 2, 2 3, 3 4, etc., as shown at Fig. 32, thereby secur- 
ing the true lengths of all lines necessary to develop the 
pattern. 

As will be noted upon examination of Fig. 33, these 
lines describe a zigzag path which crosses and recrosses 
the pattern, turning at the top of same at distances equal 
to distances between points of division of the small circle 
in plan, and at the bottom of pattern at distances equal 
to distances between points of division of the large circle 
in plan. 

Developing the Pattern. 

To develop the pattern we may draw in any convenient 
position, a line whose length is equal to line 1 1, Fig. 32. 
From the extremities of this line describe arcs whose 
radii are equal to the distances between points of division 



48 



TRIANGULATIOX 



of the large and small circle in plan, Fig. 31. With the 
compasses set to a distance equal to the length of line 
1 2, Fig. 32, place one point at point 1 of the pattern, 
(i.e., that end of line 1 1 which may be selected as the 
bottom of the pattern) and describe the small arc as at 2 
at the top, then will the intersection of the small arc 
locate the upper extremity of line 2 2. With point 2 at 
the top of pattern Fig. 33, as center, and with the length 




Fig. 34. Drawing showing how the Pattern may be Secured 
with the Least Number of Lines. 

of line 2 2 secured from Fig. 32 as radius, describe the 
small arc as shown at 2 at the bottom of the pattern. 
This, as will be noted, locates line 2 2 in its correct rela- 
tive position. 



FRUSTUM OF AN OBLIQUE CONE 49 

This completes what we may term, one section of a 
covering for the object. These sections are clearly shown 
in plan Fig. 30, and in perspective, Fig. 31, therefore 
the reader should have little difficulty in completing the 
pattern, since the same operations are involved to secure 
the true form of all sections shown in Fig. 33, although 
different lengths of lines must be employed, i. e., we must 
use the proper length for each line drawn. 

Lines drawn to connect points as 1 1, 2 2, 3 3, etc., 
have usually been drawn solid, while those drawn be- 
tween points as 1 2, 2 3, 3 4, etc., have been drawn dotted, 
the only purpose of which is to avoid confusion. 

Necessity for Lines in Pattern Development. 

In a demonstration of pattern development, lines are 
drawn to illustrate the relation between points presumed 
to be upon the surface of the object, although the demon- 
strator subjects himself to considerable criticism from 
some whose knowledge of pattern development is limited. 

The remark is frequently heard: Well, that may be 
all right, but he makes too many lines. This proves con- 
clusively that the speaker has not stopped to consider. 
Designated points and lines are as necessary in a geo- 
metrical demonstration as the letters of the alphabet are 
necessary to a printed page. 

However, when one becomes familiar with the opera- 
tions required for the solution of a problem, that problem 
may be worked out in such a manner as to appear greatly 
abbreviated as shown at Fig. 34, where the same results 
are secured as in the demonstration where Figs. 30, 31, 
32, and 33 have been shown in an endeavor to illustrate 
principles involved. 



CHAPTER VII. 
A Transitional Fitting from Oblong to Round. 




Fig. 35. A Fitting Making a Transition from Oblong 
to Round. 

Methods have been explained in foregoing Chapters 
which may be employed to secure the patterns for prac- 
tically all irregular forms whose ends are parallel. How- 
ever, for the purpose of conveying to the student an 
understanding of the application of those methods to a 
variety of forms, one or two additional examples will be 
introduced before entering into a discussion of those 
forms whose ends are not parallel. 

Fig. 35 illustrates a fitting making a transition from 
oblong to round. This, or a modification of it, is a form 

50 



OBLONG TO ROUND 51 

which is frequently demanded, i.e., we find it in many 
branches of sheet metal work, and made from all gauges 
of material. It is a fitting making a connection between 
two pipes whose axes are parallel, but not in one line. A 
change in the relative position of its ends demands little 
or no change in the methods to be employed in securing 
its patterns. We could as consistently look upon the 
oblong end as the top. It is, in a measure, a combination 
of forms which have been discussed. 

The Plan. 

The plan is secured by drawing diagrams which repre- 
sent cross sections of pipes to be connected in their 
correct relative positions. The surface is represented as 




iT' , r'T |, r i T |, r i T l T l T l T l 'r i T |, r i T , T'T l T l T l T |1 

3 6 9 IF 15 18 



Fig. 36. The Plan. 

being divided into triangles by dividing these diagrams 
into parts, and drawing lines between points of division. 
Presuming the pattern is required for a fitting as 
described above, whose diameters of oblong end are 10 
and 16 inches, diameter of round end 15 inches, and 18 
inches in hight, also making an offset of 4 inches, or, 
in other words, the round end is required to project 4 
inches beyond one end of the oblong, we would proceed 



52 TRTANGULATION 

as follows: Draw the oblong diagram to those dimen- 
sions as shown in plan, Fig. 36, in which a scale is given 
to verify measurements as given. Draw a line as 1 9 
through the points from which the semi-circles have been 
drawn which constitute the ends of the oblong. Extend 
this line from point 9 of the oblong, a distance equal to 
the required offset (4 inches), as at 9 of the circle. 

Locate a point upon line 1 9 at a distance from point 9 
of the circle equal to one half the diameter of the top 
{7y 2 inches) as at H, then will point H be a center from 
which a circle is drawn to represent a plan of the top as 
shown. Since the line 1 9 divides the plan into equal 
parts, as has been previously explained, it is only neces- 
sary to consider one part, as it may be duplicated for the 
other equal part. 

Divide the semi-circle representing one-half of the 
round end into a number of equal parts, as at 1, 2, 3, 4, 
etc., then will a point as 5 divide the semi-circle into two 
equal parts. Determine the length of the straight line 
which constitutes one side of the oblong, as between 
points 5 and 5 a of that diagram. Divide the curved por- 
tions of the oblong diagram, i.e., those semi-circles in- 
cluded between points 1 and 5, also between 5 a, and 9, 
into the same number of equal parts as each half of the 
semi-circle representing the round end has been divided 
into, as shown. Draw lines 1 1, 2 2, 3 3,4 4, 5 5, 5a 5, 6 6, 
etc., as also shown in plan, Fig. 36. These lines are now 
looked upon as plans of lines presumed to be upon the 
surface of the object. 

As will be noted, the above lines are not sufficient to 
represent the surface as being divided into triangles; 
However, upon drawing lines as 1 2, 2 3, 3 4,4 5, 5a 6, 
6 7 , etc., we find the whole surface of one-half of the 
object represented has been divided into triangles. 



OBLONG TO ROUND 



53 



Right Angled Triangles. 

Our next operation is to determine the true lengths of 
these lines, or the true distances hetween designated 
points of the base and top. This is accomplished by the 
use of the right angled triangle, as has been frequently 
explained, and here shown at Fig. 37, where, to avoid 
confusion two sets are employed. 



\v \ \ \ W4-5 







\ \-\ 



6-7 

7-8 
8-9 



\ \ 



Fig. 37. Diagram of Triangles. 

As will be noted, the perpendiculars of all triangles, 
Fig. 37, are equal to the vertical hight of the object, or 
18 inches, as shown at A B and D E. The bases of said 
triangles being equal to lengths of lines in plan, as shown 
in spaces from B and E along lines B C and E F, thus 
locating points between which lines may be drawn to 
secure the true lengths of those lines shown in plan which 
divide the surface of the object into triangles, and con- 
nect points of the base with points of the top. 



The Pattern. 

As has been explained, we may look upon the lengths 
of lines in the diagram of triangles as the distances be- 
tween points of the base, and points of the top. Since the 



.54 



TRIANGULATION 



plan supplies the distances between these points along 
the base and top, all necessary measurements are before 
us to develop the pattern, and the mental process may 
run through our minds somewhat as follows : Draw in 
any convenient position upon the plane of development, 
a line whose length is equal to the length of line 1 1 found 
in the diagram of triangles, Fig. 37. Decide which end 
of this line shall be at the base of the fitting and mark 
accordingly, as shown. The distances between points 1 




Fig. 38. The Pattern for One-half of the 
Fitting. • 



and 2 at the base and top of the object are shown in 
plan, therefore we describe arcs with these distances as 
radii, and with points 1 and 1 of the pattern as centers, 
when points 2 and 2 must lie in these arcs. 

We have the true distance between points 1 of the base 
and 2 of the top, in the length of line 1 2 of the diagram 
of triangles, therefore if we describe an arc with this 
radius, and point 1 at the bottom of the pattern as center, 



OBLONG TO ROUND 55 

its intersection with the first small arc at 2 must be the 
exact location of point 2 at the top of the fitting, or 
pattern. 

With the compasses set to a span equal to the length of 
line 2 2 found in the diagram of triangles, and with point 

2 at the top of the pattern as center, describe the second 
small arc at the bottom of the pattern when the line 2 2 
may be drawn, thereby completing what may be looked 
upon as one section of the pattern or covering of the 
object, as shown in plan Fig. 36, at 1 2 2 1. 

The next step is to draw two additional arcs with 
points 2 and 2 of the pattern as centers, and with dis- 
tances as found between points 2 and 3 of the plan as 
radii. The distance between point 2 at the base and 
point 3 at the top of the object is shown in the length of 
line 2 3 of the diagram of triangles, which we may use 
as before to secure the exact location of point 3 at the 
top. In a similar manner we use the length of line 3 3, 
found in the diagram of triangles, to locate point 3 at the 
bottom of the pattern. 

Two more small arcs are added as before, using points 

3 3 of the pattern as centers, when points 4 4 of the top 
and base may be located by transferring the distances as 
found in the lengths of lines 3 4 and 4 4 of the diagram 
of triangles. In the same general manner as has been ex- 
plained, points 5 5 of the pattern may be located as 
shown. 

We find, upon referring to the plan, that there are two 
lines radiating from point 5 of the circle, which are the 
plans of lines which connect point 5 of the top to 5 at the 
base, also 5 at the top with 5a at the base, and including 
two sides of a flat triangular surface upon the side of the 
object. The diagram of triangles supplies in the length 



56 TRTANGULATION 

of line 5a 5, the distance from point 5 to 5a upon the 
surface of the object, therefore we may use that distance 
as radius, with point 5 at the top of the pattern as center, 
to describe a small arc as shown at 5a. The plan 
supplies the distance from point 5 to 5a at the base 
of the object, and using that distance as radius, with 
point 5 at the bottom of the pattern as center, we may 
describe an arc, cutting the first at 5a, thereby locating 
point 5a upon the pattern in its correct relative position. 
Since the remaining points shown upon the pattern are 
located in the same manner as heretofore explained, it 
seems that one will have little difficulty in completing the 
pattern as shown. Those who have given this work 
the attention that the subject demands, beginning with 
the first Chapter, should now be in a position to develop 
the patterns for a variety of forms whose ends are 
parallel, although some care must be exercised when de- 
signing them. 

Forms of Fittings. 

When the centers of the ends are approximately in one 
line which is perpendicular to the planes of said ends, the 
fitting may be made comparatively short. However, in 
every instance a moderate length is more convenient 
since the change of form is less per unit of length, there- 
fore the metal responds more readily when forming it 
to its required shape. When the offset is considerable, 
and it is desirable to preserve the capacity of the fitting, 
i.e., the area of its cross section, it becomes necessary to 
increase its length. For example, Fig. 39 illustrates 
transitional offsets, or connections between round and 
rectangular pipes, where the offset is as shown. If the 
fitting is made as shown at A, its capacity at a b will be 
considerably reduced. However, if its length is in- 



OHLONG TO ROUND 



57 



creased as shown at B, its capacity at a b will be corre- 
spondingly increased. Conditions will not at all times 
permit of this increase in length, therefore we must re- 
sort to a form of fitting as shown at C. Here, as will be 
noted, that portion which forms the transition is con- 




Fig. 39. Forms of Fittings. 

nected to pipes which have been cut obliquely, and in 
many instances the ends of the center portion will not be 
parallel. To develop patterns of this class demands a 
somewhat greater understanding of the science, and is 
explained in subsequent Chapters. 



CHAPTER VIII. 

A Two Pronged Fitting Which Can Be Made in 

One Piece. 




Fig. 40. Pictorial View of a Two-pronged Fork. 

A two pronged fitting as illustrated at Fig. 40, made 
from one piece, supplies an interesting and instructive 
example in pattern development. 

The student's attention is directed to this as one worthy 
of careful attention, since a clear understanding of the 
positions of triangles which must be presumed to com- 

58 



AN [NTERESTING EXAMPLE 



59 



pose its surface when the pattern is developed, will with- 
out question, advance one's understanding of other forms 
which will be encountered. 

Many modifications may be made of it without ma- 
terially changing the methods of securing its pattern, 
providing its ends remain parallel. 




Fig. 41. Pictorial View of the Fitting, Showing Triangles 
Presumed to be Upon its Surface. 

There is, perhaps, no ironclad rule which must be fol- 
lowed in locating the above spoken of triangles, although 
they should be so located as to allow all lines which are 
the boundaries of said triangles to be as nearly straight 
lines as possible when placed upon the surface of the 
object. 

Some judgment will also be necessary to determine 
what amount of the collar here shown as the top shall 
be devoted to each prong of the fitting. Fig. 41 shows 



60 TRIANGULATION 

the body of the object in a pictorial way, and the tri- 
angles the author has presumed to be upon its surface. 

Upon giving Fig. 41 attention, the student will note 
that the body of the object is composed of parts of oblique 
cones. As for example, where a number of full lines 
radiate from a single point, that portion of its surface 
included within those lines is a portion of an oblique 
cone, and that portion of its surface where broken lines 
are shown which alternate the full lines, is a portion of 
the frustum of an oblique cone. 

Therefore, as above stated, the whole surface is com- 
posed of portions of oblique cones whose bases and ver- 
tical hights are of varying dimensions. 

On the Characters Used in Pattern 
Demonstrations. 

The multiplication of characters to designate similar 
points in different diagrams employed to solve a problem 
in pattern development, is always a source of annoyance. 
Therefore in an endeavor to reach the reader's mind 
directly through the medium of the eye, the author has 
made it an almost universal rule to designate similar 
points by the same character in each diagram. As, for 
example, points B and 2, Fig. 41, are the upper and lower 
extremities of line B 2, and this line is shown in plan, 
Fig. 42, between points B and 2. In the diagram of tri- 
angles line B 2 is shown in its true length, and designated 
as B 2. 

The pattern shows line B 2 in its correct relative posi- 
tion, with one extremity at the top, while the other is at 
the bottom, and if the pattern be wrapped about the 
object as shown at Fig. 41 in a manner as to allow line 
B 2 to coincide with line B 2 there shown, all other desig- 
nated points or lines must also coincide. 



AN INTERESTING EXAMPLE 61 

On the Plan. 

Since the plan of an object as here illustrated may be 
divided into two equal parts, one part as shown at Fig. 
42 will fulfil every requirement in developing its pattern. 
On the other hand, it may in some instances be advisable 
to draw a complete plan, for the purpose of securing a 
clea r er understanding of the object and its surface. This 
must be done if it is required to secure the pattern for 
an object whose surface cannot be divided into equal 
parts. 

It may be here explained that to secure the pattern for 
an object whose surface cannot be divided into equal 
parts, the whole surface must be represented. This does 
not imply that there are any additional principles to be 
applied, but simply that there is an increased number of 
lines whose lengths and positions must be determined. 

If the pattern cutter has any difficulty in securing the 
pattern for an object whose surface cannot be divided 
into equal parts, it is very likely due to his inability to 
form a clear conception of the object. One who finds 
himself thus handicapped should devote some time to 
the study of the relation the plan bears to the object, and 
remember, as stated above, that there are no additional 
principles involved, simplv an increased number of lines 
to be dealt with. 

The First Step to Secure the Pattern. 

The first step to secure the pattern for an object as 
illustrated at Fig. 40 is to draw a plan, or a portion of it. 
This, as has been previously stated, should be as simple 
as the nature of the object will permit. Since the fitting 
is designed to make connection between three round pipes 
whose axes are parallel, we may presume to view these 



62 TRIANGULATION 

pipes from above as in orthographic projection, and draw 
the circles in plan which shall represent the cross-sections 
of said pipes as shown at Fig. 42, assuming the diameters 
of these pipes to be as shown. Here the semi-circle A E J 
is a plan of one-half of the top, and the small semi-circles 
constitute a plan of one-half of the base. 

The next step is to determine what amount of the 
larger semi-circle shall be devoted to each small semi- 
circle, or, in other words, what part of the top shall be 
devoted to each of the small collars. In this instance, five- 
eighths of the arc A E J has been devoted to the large 
collar at the base, and three-eighths to the small one, i.e., 
the point F is connected by lines to each small semi- 
circle. 



; f"- \ A 


•t- - 


~ 9jt~\jo j r^-- j 78 


jbr—^~~- \ „ 




8 1\k n 4* — "v" 


v^t — V 




A \ v* 7 "^/ 


\ V ^ \ 




7/ \\\ \Hl%^^ 16 






3 N^C^ 




j^\\l Z^r 1 * 


4^ 


1>^ 






Fig 


. 42. Semi-plan. 



We may now divide each of the semi-circles into an 
equal number of equal parts as shown from 1 to 9, and 
from 10 to 18. If each semi-circle is divided into eight 
parts, as here shown, the point F is located without 
further trouble, thereby locating six points as A, B, C, D, 
E and F , which may be connected to similar points of 
the base as shown by lines 1A, 2B, 3C, 4D } 5E, and 6F. 
These lines may now be looked upon as being elements 
of the surface of the frustum of an oblique cone, and the 
pattern for that portion may be secured in the same man- 
ner as explained for that form in Chapter VI. We may 
now connect points 7 , 8 and 9 to F, and look upon that 
portion of the object as a portion of an oblique cone. The 



AN INTERESTING EXAMPLE 



63 



surface represented in plan within the triangle 9 10 F is 
a flat surface. 

That portion of the large semi-circle between F and / 
may now be divided into one-half the number of equal 
parts that the semi-circle 10,14,18 has been divided into 
(in this case four), and lines drawn as F 14, G 15, H 16, 
1 17, and J 18. The remaining points as 10, 11, 12 and 
13 may be connected to F, thereby securing the plans of 
lines which are presumed to be upon the surface of one- 
half the object, and shown in a pictorial way at Fig. 41 . 



Triangles. 



The lengths of the above spoken of lines are now em- 
ployed as the bases of triangles whose perpendiculars are 




\\\ H£v 



B-l 
C-2 
0-3 



W\ '. \ \\\\ S-14 



TV7X 



H-15 
-1-16 
-J-17 



Fig. 43. Diagram of Triangles. 

equal to the vertical night of the fitting as shown at Fig. 
43, where K L is presumed to be equal to that hight. The 
length of lines in plan are set off from L along line L M, 
and these points connected to K, thereby securing the 
true lengths of all full lines shown in plan. 

Since portions of the fitting are parts of the frustums 



64 



TRIANGULATTON 



of oblique cones, some additional lines must be assumed 
to completely divide the surface into triangles. These 
lines are shown in plan, and in Fig. 43 as broken lines, 
and their lengths are secured in the same general manner 
as has been explained and shown at Fig. 43. 

The Pattern. 

Having before us the true lengths of all lines necessary 
to develop the pattern, we may proceed by drawing in 




Fig. 44. The Semi-pattern. 



any convenient position upon the plane of development, 
a line whose length is equal to the length of line A 1, 
Fig. 43, as shown at A 1, Fig. 44. The distance from 
A to B is found in plan, and the true distance from 1 to 



AN INTERESTING EXAMPLE 65 

B is the length of line B 1, Fig. 43, thus enabling us to 
locate point B in its correct relative position upon the 
plane of development, as shown at Fig. 44. 

Point 2 may now be located, since the plan supplies 
the true distance from 1 to 2, and the diagram of tri- 
angles supplies in line B 2, the true distance from B 
to 2. Points C, D, E and F at the top, and 3, 4, 5 and 
6 at the base, may all be located In the same general 
manner. 

Presuming that line F 6 has now been located upon the 
plane of development, a glance at the diagrams will 
show that lines F 6 to F 14 inclusive, all radiate from 
point F. 

The diagram of triangles supplies the true lengths of 
these lines, and the plan supplies the true distances said 
lines are from each other at their extremities, therefore 
little trouble should be experienced in locating points 6 
to 14 upon the bottom of the pattern as shown. Since 
the remainder of the required semi-pattern, or points G, 
H, I and J, also 15, 16,17 and 18, are located in the same 
general manner as were similar points shown at the left 
side of Fig. 41, the reader should have little difficulty in 
completing the work as shown. 

It may be remarked that slightly more accuracy may 
be obtained by first locating upon the plane of develop- 
ment, that surface within the triangle F 10 9, and then 
adding the triangles at each side of this. This, as will 
be noted, eliminates some opportunity for error which 
may have been committed in the early part of developing 
the pattern. However, this is a matter for the operator 
to decide, since if care be used the difference will be 
slight. 



66 triangulation 

When it is Required to Fit the Ends of the Object 

to Round Collars Whose Circumferences 

Have Been Established. 

There is a constant ratio between the circumference of 
a circle, and its diameter, the value of this ratio to six 
figures is 3.14159; however, for all ordinary purposes, 
3.14 is sufficiently accurate; therefore we may determine 
the diameter of any circle whose circumference is given, 
or, we may determine the circumference of any circle 
whose diameter is given, by either multiplying or divid- 
ing as the case may require. As for example, diameter 
multiplied by 3.14 equals the circumference, or the cir- 
cumference divided by 3.14 equals the diameter. 

When the circle is drawn and divided into a number 
of equal parts, for example, twenty-four, each part repre- 
sents one twenty-fourth of its circumference, and as ordi- 
narily measured each space is a straight line, or the chord 
of an arc. Since the chord is always less than the arc it 
subtends, the twenty-four spaces along a right line will 
very likely be something less than the figured circum- 
ference, thereby introducing some error. Therefore, if 
the circles be drawn in plan as accurately as may be, and 
their known circumferences set off upon right lines, these 
lines may be divided into the same number of equal parts 
as the circles have been divided into, and these spaces 
upon right lines employed as the correct distances to be 
set off upon the pattern, some more accuracy may be 
obtained. 



CHAPTER IX. 

Some Principles of Orthographic Projection as 
Applied to Triangulation. 



To secure the pattern for an object whose ends are not 
in parallel planes demands a greater knowledge of ortho- 
graphic projection. The reader will note that in fore- 
going examples a plan of the object, together with a 
knowledge of its vertical bight was sufficient to enable 
us to determine the true lengths of all lines presumed to 
be upon its surface and shown in plan. The reason for 
this is found in the fact that all points of the top are the 
same vertical distance from the plane of the base, or, all 
points of the base are at the same vertical distance from 
the plane of the top. 




Fig. 45. A Pictorial View of Transition Pieces and the Planes 
Within which their Ends arc Situated. 

When the ends of the object are not parallel a more 
complicated problem is encountered, since there is varia- 
tion in the distances between the planes of its ends. 
Therefore some method must be employed which will 

67 



68 TRIANGULATION 

enable us to determine the distance between different 
points which may be conceived as being located within 
those planes. The above is clearly shown at Nos. 1 and 2, 
Fig. 45. 

Fig. 45 shows, in a pictorial way, objects making a 
transition from square to octagonal, and the planes A B 
C D, within which the ends of those objects are situated. 
It is apparent upon examination of No. 2, that the per- 
pendiculars of triangles employed to secure the true 
lengths of lines, must be of varying lengths. In an en- 
deavor to convey to the reader an understanding of the 
principles involved to secure those lengths, some ele- 
mentary discussion relating to the point, right line, and 
plane will be introduced. 

On the Representation of a Point Upon the 
Vertical and Horizontal Planes of Projection. 

Since the relative positions of points, lines and planes 
must be determined when the solution of the more com- 
plex problems in pattern development are attempted, we 
shall first consider the surfaces upon which they are 
represented. As has been previously explained, a plan 
is usually drawn upon a surface which is presumed to 
be horizontal. An elevation can be, and is many times, 
drawn upon the same surface, but intended to represent 
the object when viewed from positions which are at right 
angles to those assumed for the plan. Therefore if the 
object is to remain stationary, the surfaces upon which 
the plan and elevation are to be drawn must be presumed 
to be at right angles to each other. Thus we have what 
are known as the vertical and horizontal planes of pro- 
jection. 

As for example, the surface ABC D, No. 1, Fig. 46, 



ORTHOGRAPHIC PROJECTION 



69 



represents the surface upon which a plan and an eleva- 
tion may be drawn. A line, real or assumed as / L, 
divides this surface into convenient parts. The lower 
portion as I L C D remains in a horizontal position, 
while the upper portion as A B I L is looked upon as 








No. 3 






A 


h 


b 


B 






S2 


Lt 








1 




* 


L 


1 




i 


Pj 



No. 4 



Hin. 



10 in. 



No.5 



No. 6 




2nd. 



3rd. 



1st. Angle 



4th 



No.7 No.8 

Fig. 46. Illustrating the Principal Planes of Projection. 



being in a vertical position, as illustrated at No. 2, Fig. 
46. That portion of the surface as shown at / L C D is 
known as the horizontal plane of projection, and that 
shown at A B I L as the vertical plane of projection. The 
line I L is the intersecting line between the two planes. 



70 TRiANGULATION 

As these planes are presumed to be capable of indefinite 
extension there is no limit as to size. 

The above spoken of planes, i.e., the vertical and hori- 
zontal, are known as the principal planes of projection, 
and are sufficient for many, but by no means all, of the 
ordinary operations of pattern development. In addition 
to the above there are the profile and oblique planes, 
which must be employed at times to secure desired re- 
sults. However, since a knowledge of the first is essential 
to the study of the others, the author will for a time con- 
fine himself to the representation of the point, right line, 
and plane upon the vertical and horizontal planes of 
projection, and, as the work progresses, endeavor to 
explain the positions and value of the profile and oblique 
planes. 

The object to be represented is presumed to occupy a 
position in space above the horizontal, and in front of 
the vertical plane. It may be here explained that space 
is unlimited extension in which all bodies are situated. 
The absolute position of bodies or objects cannot be 
designated except in a relative way, i.e., by referring 
them to each other, or to objects whose positions are 
assumed. In orthographic projection all objects are re- 
ferred to the planes of projection. 

Since representations of objects upon the planes of 
projection are composed of lines, and as lines are made 
up of points, we may direct our attention for the moment 
to the projection of a single point. 

The point, which is the least of geometrical magni- 
tudes, if considered as a visible particle, can be located in 
space by giving its distance from each of the two prin- 
cipal planes of projection. 

Presuming a point is located 14 inches above the hori- 
zontal plane, and 10 inches in front of the vertical plane 



ORTHOGRAPHIC PROJECTION 71 

as shown at No. 3, Fig. 46, then a pictorial view of the 
planes and point as at E is shown at No. 4, Fig. 46. If 
from point E in space, a perpendicular line be let fall to 
the horizontal plane, the foot of the perpendicular as G 
is the horizontal projection or plan of the point. If in like 
manner, a perpendicular be drawn to the vertical plane, 
the point of intersection with that plane, as at F, is the 
vertical projection of the point, or its elevation. These 
perpendiculars are called the projecting lines of the point. 

It will be noted that this places the plan of the point 
at the same distance from /Las said point is known to 
be from the vertical plane, and the position of its eleva- 
tion is at the same distance from / L as the point is 
known to be above the horizontal plane. The converse 
of this may be assumed, i.e., the location of the point in 
space is determined by its projection upon the vertical 
and horizontal planes, since its plan is 10 inches in front 
of the line / L, the point itself must be 10 inches in front 
of the vertical plane, and since the elevation of the point 
is 14 inches above / L, the point itself must be 14 inches 
above the horizontal plane. If a plane which is perpen- 
dicular to the two planes of projection be passed through 
point E in space, as shown atabc d, No. 5, Fig. 46, said 
plane would cut a right line from each, i.e., the vertical 
and horizontal planes of projection, as illustrated by lines 
b d and d c, No. 5, Fig. 46, which are at right angles to 
/ L. Therefore the elevation of a point will be found in 
a line drawn from the plan of said point perpendicular 
to / L, or the plan of a point will be found in a line let 
fall from the elevation of said point and perpendicular to 
/ L when the vertical plane has been so revolved as to be 
parallel to the horizontal as shown at No. 6, Fig. 46. 

In other words, the plan and elevation of a point are 
found in a right line drawn perpendicular to / L. Its 



72 TRIANGULATION 

distance above line I L is equal to the distance said point 
is above the horizontal plane, and its distance below line 
I L is equal to the distance said point is in front of the 
vertical plane. This, as will be noted, places the eleva- 
tion above the plan in every instance. Thus we have 
what is commonly known as a first angle projection. 

On the Relative Positions of the Plan and 
Elevation. 

There is a tendency among draftsmen to place their 
elevation below the plan, or, in many cases, in what seems 
to be the most convenient position for them at the 
moment. This the writer believes is more likely to con- 
fuse than enlighten. Geometrical authorities state that 
the first angle is sufficient for all ordinary operations, and 
as it is by far the most simple of comprehension, the 
author will in every instance locate his object in the first 
angle. This is in line with the teachings of an English 
instructor in civil engineering, and a writer on ortho- 
graphic projection who has always been held in high 
esteem. 

In explanation of the above it may be stated that 
geometry teaches that the intersecting line is not a limit- 
ing line, but the line where two planes which are capable 
of indefinite extension intersect or cross one another, as 
shown at No. 7, Fig. 46. This forms four equal angles 
as shown at No. 8, Fig. 46, where an edge view of the 
planes is shown. The object may be looked upon as 
being situated within either of these angles. Thus when 
the vertical plane is presumed to be revolved into the 
plane of the paper, the elevation will occupy a position 
dependent upon the angle within which the object is 
situated. This then explains to some extent, the different 
positions taken for plans, elevations, and sections. 



ORTHOGRAPHIC PROJECTION 7$ 

Geometrical authorities also state that the point of 
sight is always at an infinite distance above the hori- 
zontal and in front of the vertical plane, which is within 
the first angle, hence all objects situated within this angle 
can be seen. Objects situated within either of the other 
angles are concealed more or less by the planes of pro- 
jection. Lines that are given or required are made full 
if they can be seen, but are dotted if concealed by other 
objects, or by the planes of projection. Auxiliary lines, 
or lines used to aid in the construction of a problem, are 
always dotted. 

Many broken or dotted lines found in a pattern demon- 
stration are included simply as an aid in conveying an 
understanding of the problem, although it must be ad- 
mitted that in many instances said lines are erroneously 
looked upon by the novice as confusing the demonstra- 
tion. 



CHAPTER X. 

The Representations of Objects on the Vertical, 
Horizontal, Profile and Oblique Supplemen- 
tary Planes of Projection. 

Having explained in Chapter IX the principles in- 
volved in the representation of a single point upon the 
vertical and horizontal planes, attention will now be 
directed to the representation of that solid known as a 




Fig. 47. A Pictorial View of the Vertical and 
Horizontal Planes, together with a Cube Located 
Within the First Angle. 

cube upon these planes. It will be remembered that a 
cube is a solid bounded by six equal faces or squares and 
having all its angles right angles. 

Fig. 47 is a pictorial view of the vertical and hori- 
zontal planes in their assumed positions, with the cube 
suspended in space in front of the vertical and above the 
horizontal plane, with two of its faces parallel to each. 
If the cube be viewed from above, with the point of sight 

74 



PLANES OF PROJECTION 



75 



moving over the object so as to place every point viewed 
in a line perpendicular to the horizontal plane of projec- 
tion, a square equal to one of its faces would represent 
it as shown at a b c and d, Fig. 48. 

Perhaps this will be more fully comprehended if we 
presume to drop plumb lines from the vertice of angles 
E F G and H, Fig. 47, to intersect the horizontal plane 



SleuatiQJi 




Fig. 48. The Plan and 
Elevation of a Cube. 

in points m p and /, where lines are drawn to connect 
them, thus forming a square equal to one face of the 
cube. It should be noted that point m is not only the plan 
of point G, but the plan of point M as well, also the plan 
of line G M } or any number of points along the line G M. 
This deduction is not confined to the line G M, but can be 
applied to all lines which are perpendicular to the hori- 
zontal plane of projection. Line m p, Fig. 47, is not only 
the plan of line G H, but of the line M N also, or a plan 
of any line which may be drawn upon the surface G H 
N M, and intersecting lines G M and H A r . 



76 TRIANGULATION 

Thus it will be noted that a point in plan may represent 
a point in space, or a line which is perpendicular to the 
horizontal plane of projection. Likewise a line in plan 
may represent a line in space either parallel or oblique to 
the horizontal plane, or it may represent a plane which is 
perpendicular to the horizontal plane of projection. 

The elevation of the object is secured by drawing a 
square equal to one of its faces, directly above the square 
which is looked upon as a plan, as shown at e f g and h, 
Fig. 48. This is also illustrated at Fig. 47, where the 
lines G E,H F, and TV if produced, would intersect the 
vertical plane A B I Lin points g h and n, thus locating 
points which may be connected to form an elevation as 
shown at g h and n. Here the point g is not only the 
elevation of points G and E, but of any number of points 
along the line G E, likewise the point h is an elevation 
of points H and F, or of any number of points along the 
line H F. Similar conclusions may also be drawn for the 
line N 0. The line g h is not only an elevation of G H, 
but the elevation of line E F as well, and line g h is also an 
elevation of the face E F H G, or of any line which may 
be drawn upon that surface which intersects lines E G 
and F H. Therefore a line in elevation may represent a 
line in space which is parallel or oblique to the vertical 
plane of projection, or it may represent a plane which is 
perpendicular to the vertical plane of projection. 

In proof of the above, we may presume to draw a line 
from G to F upon the upper face of the cube, Fig. 47. It 
will then be noted that the line g h is its elevation, and a 
line drawn from m to / would then be its plan, thus show- 
ing that this line is oblique to the vertical plane of pro- 
jection. 

It should be remembered that Fig. 47 is a pictorial view 
of the cube and planes of projection, which is introduced 



PLANES OF PROJECTION 77 

for the purpose of giving a clearer understanding of the 
principles involved, while the plan and elevation proper is 
at Fig. 48, presuming the cube to be of a size as shown. 

If the reader has become sufficiently interested he will 
do well to provide himself with a drawing board and a 
few accessories, which he may obtain from almost any 
dealer in artist's materials. The drawing board should be 
of convenient size, say 23 x 31 inches. He will also re- 
quire a T-square which has a blade about equal in length 
to the drawing board ; two triangles, one 8 inch 45 de- 
grees, and one 10 inch 60 and 30 degrees, a few thumb 
tacks, a lead pencil and rubber. Almost any grade of 
paper may be used. In regard to drawing instruments, a 
pair or two of compasses will be all that is required. 

If the student will perform a few experiments which 
the foregoing should suggest, by the aid of pieces of card- 
board, and remember that the plan of a point or line 
will always be found directly beneath it, and that the 
elevation is always found directly back of it, the above 
will no doubt be made quite clear, and place him in a 
position to grasp additional facts which will aid him as a 
pattern cutter. 

The Profile Plane. 

The profile plane is an additional plane assumed to be 
perpendicular to the principal planes of projection, i.e., 
the vertical and horizontal. No. 1, Fig. 49 illustrates at 
A B LI the vertical plane, and at / L D C the horizontal 
plane, then a b c d is a profile plane, and a representation 
upon this plane is termed a profile. The profile plane may 
be presumed to be revolved about a perpendicular axis 
as the line b c, into the plane of the paper, or about a 
horizontal axis as the line c d, and may be located either 
to the right or left of the assumed position of the object. 



78 



TRIANGULATION 



The line about which the profile plane is presumed to be 
revolved becomes an intersecting line" when the projec- 
tion upon that plane is constructed' in the same general 
manner as has been explained for projection upon the 
principal planes. 

The profile plane is often of the greatest use, which is 




Na.3 



No 4 



Fig. 49. Illustrating the Profile and Oblique Supplementary 
Planes. 

illustrated in a simple example at No. 2, Fig. 49. A tri- 
angle as E F G is presumed to be suspended in space so 
that its surface is perpendicular to the vertical and hori- 
zontal planes of projection. The line / h is a plan of 
said triangle, and the line e g is its elevation. To secure 
the true form of the triangle or the length of its longest 
side, it must be revolved until it becomes parallel to the 
plane upon which it is represented, or an additional plane 
assumed, which is here shown as the profile plane abed. 



PLANES OF PROJECTION 79 

This plane being parallel to each of the three sides of the 
triangle, its representation upon that plane will be its 
true form, as shown in a pictorial way at S T U. 

Supplementary Oblique Planes. 

It is frequently found convenient to make use of what 
is known as oblique planes. The object is represented 
upon these, its position being fixed as regards the princi- 
pal planes. The positions of such planes are determined 
by conditions of convenience, and therefore depend upon 
the nature of the object, but they are, in most cases, such 
that these planes are perpendicular to one of the principal 
planes. The oblique plane is shown pictorially at No. 3, 
Fig. 49. For example, A B L is the vertical and C D L 
the horizontal plane of projection, then a b c d is the 
oblique plane. 

To illustrate the use of the oblique plane, let it be 
presumed that the surface A B D C oi No. 4 in Fig. 49, 
is a surface upon which a right angled triangle is to be 
represented in plan and elevation, when its position is as 
follows: The surface of the triangle is at an angle of 
60 degrees to the horizontal plane, with its longest side 
parallel to the vertical plane. If a line as X Y be drawn 
at an angle of 60 degrees to the line / L, this line, i.e., 
X Y, may for the moment be looked upon as the inter- 
secting line between the vertical and a supplementary 
plane which makes an angle of 60 degrees to the hori- 
zontal plane, then draw the triangle upon this plane in 
its true form, keeping its longest side parallel to line X Y 
as shown at a b c. Draw upon the vertical plane to the 
right of line X Y a line as V a' c' parallel to line X Y. 
Project points a b c as shown at V a' c f , then will the line 
V a' c' be a representation of the triangle upon the verti- 
cal plane. 



80 TRIANGULATION 

Its plan is secured by dropping lines from points b f a' c r 
perpendicular to / L, and locating" points upon these lines 
at distances from / L as found from the line X Y to 
points a b c, as shown at b" c" a" , after which lines are 
drawn to complete the plan as shown. It will be noted that 
the line b c as shown in plan at b" c" is considerably fore- 
shortened for the reason that this line as at an angle to 
the horizontal plane of projection. 

As has been previously stated, the principles employed 
to secure patterns for forms whose ends are not parallel 
are somewhat complex. On the other hand, if one 
secures a clear understanding of the principles employed 
to secure the patterns for one form, he may employ those 
principles for all. He who considers the art of pattern 
cutting worthy of attention will find that a study of its 
principles is of the utmost importance. 

The work may here seem to be somewhat extended; 
however, the student is cautioned not to turn it aside 
and wait for something which seems for the moment to 
be more in the line of this work, since, if the principles 
as herein explained are not fully comprehended by the 
pattern cutter, he can never become a master of his art. 
The study of individual pattern demonstrations, and the 
pursuit of this or that one's methods may enable one 
to develop a considerable number of patterns, but at 
what moment something which we have never seen may 
come before us no one can foretell. There are but few 
principles to be understood which may be applied to those 
forms which are secured by triangulation, and since the 
writer has undertaken the task of explaining those prin- 
ciples, he trusts that the student will withhold judgment 
until he has given the work considerable attention, thus 
placing himself in a position to comprehend the principles 
involved in the pattern problems which follow. 



CHAPTER XL 

The Pattern for a Fitting Whose Ends Are Not in 
Parallel Planes. 

The difference in appearance of diagrams used by dif- 
ferent operators to secure the pattern for one and the 
same object is due absolutely to the position each in- 




Fig. 50. Photographic View of a Fitting Whose Ends Are 
Not in Parallel Planes. 

dividual assumes the object to occupy as regards the 
planes of projection. It is hardly to be expected that all 
will conceive it as being in the same position, therefore 
there is variation. 

81 



82 TRIANGULATION 

In looking over a considerable number of demonstra- 
tions, the writer finds but slight if any reference to the 
planes of projection. As has been stated, an object can 
only be located as regards the planes of projection; this 
being true, it would seem important that the pattern cut- 
ter should understand the use and value of said planes. 

Upon referring to Fig. 51, the reader will note three 
pictorial representations of the irregular portion of one 
form as illustrated at Fig. 50, but occupying different 
positions as regards the surface upon which said form is 
represented as resting. 

When the pattern is required for a fitting of this class, 
at least two views will be necessary to enable us to de- 
termine the true lengths of lines presumed to be upon its 
surface. To simplify the problem rests upon our ability 
to assume the object to be in such positions as to allow the 
simplest of these diagrams to represent it in plan and 
elevation. 

Should we assume the object to occupy a position as 
shown at A, Fig. 51, the plan of the top will then be an 
ellipse. To draw this ellipse in its correct form and 
dimensions, we may assume a supplementary oblique 
plane, which is not a particularly difficult operation. 
However, since this involves the use of an additional 
plane, perhaps the problem will be simplified by revolving 
it about a horizontal axis until the plane of the round 
end becomes parallel to the horizontal plane of projection, 
as shown at either B or C, Fig. 51. 

It may be here explained that the plan of the object 
when presumed to be in positions as shown at B or C, 
Fig. 51, will be similar, and that the only variation which 
will exist will be in the position of the elevation as re- 
gards the intersecting line. 



ENDS NOT IN PARALLEL PLANES 83 

In an endeavor to explain the principles involved in 
problems of this nature, a position of the object as shown 
at C, Fig. 51, is first assumed. As a second example, the 
same object will be assumed to occupy a position as shown 
at A, Fig. 51, and the plan and elevation drawn, with its 
pattern developed, thereby illustrating the use of the 



B 



\ \ 


\ \ 


i\ 


\ 


\\ 


c 


W 




I 

1 V 


/'' 



Fig. 51. Sccnographic Representation of One Irregular Form 
Occupying Different Positions. 

oblique supplementary plane. This will clearly show how 
two patterns which finish the same may be developed from 
two sets of diagrams presenting a wide variation in ap- 
pearance. These diagrams are, in fact, the representa- 
tions of the original object, which simply occupies dif- 
ferent positions as regards the principal planes of pro- 
jection. 



84 TRIANGULATION 

To Develop the Pattern for an Object as Illus- 
trated at Fig. 50, Presumed to Be in a Position 
as Shown at C, Fig. 51. 

For convenience in verifying measurements, a scale 
has been included in Fig. 52. It has been presumed that 
the object has dimensions as follows : base, square with a 
length of side of 16 inches; top, round and 14 inches in 
diameter ; vertical hight of the shortest side, 8}i inches ; 
overhang of the shortest side, 3 inches; inclination of 
the planes within which the top and base are presumed 
to be situated, 45 degrees. 

The first step is to draw diagrams which will correctly 
represent the object in plan and elevation. In this in- 
stance the elevation is first drawn, since it may be com- 
pleted without reference to other diagrams. It is in 
reality a section of the object. 

To secure this elevation, we may draw at a suitable 
distance above the line I L, Fig. 52, and parallel thereto, 
an indefinite right line as 1 9 F. From some point upon 
this line, as at in, draw a line at an angle of 45 degrees to 
/ L, as A B in, then will the elevations of the top and 
base of the object lie in some portions of these lines. 

Since the vertical hight is 8% inches upon the shortest 
side, we may employ the steel square to locate point 9 
of the top by allowing one edge of the blade to lie parallel 
to the line A B in, and with the 8>Y\ inch mark of the 
tongue intersecting the line 1 9 in, draw a line as 9 k. As 
the top overhangs 3 inches upon this side, we lay off 3 
inches from point k as shown at B. We are now enabled 
to locate point A, since we know it must be 16 inches 
from B along line A B, and as the top is 14 inches in 
diameter, we can also locate point 1 in like manner. Upon 
drawing lines A 1 and 9 B, the elevation is completed. 



ENDS NOT IN PARALLEL PLANES 



85 




(i, 



Oh 



t-H 



tq 



t>0 



86 triangulation 

The Plan. 

The top is round and parallel to the horizontal plane, 
therefore a circle whose diameter is equal to the diameter 
of the top will represent that end in plan, its position 
being determined as clearly shown by construction lines. 
The object being composed of equal and opposite halves, 
we shall only concern ourselves with one half, or that 
portion of the plan shown below line 1 X 9 Y. 

Two sides of the base may be represented in plan by 
drawing lines from points A and B of the elevation per- 
pendicular to the line / L. In other words, points A and 
B of the elevation are in reality the end elevations of 
lines which are perpendicular to the vertical plane of 
projection, and as the specification supplies their length, 
i.e., 16 inches, it only becomes a question of locating their 
extremities in plan. Since the line 1 X 9 Y represents 
the center of the object, these lines may be made 8 inches 
in length from points X and Y , thereby locating points 
A and B upon the horizontal plane as shown in plan, 
where line A B is drawn to complete this view. 

Locating Lines Which Divide the Surface of the 
Object Into Triangles. 

With the plan and elevation completed, we are now in 
a position to locate lines which will divide the surface into 
triangles. This may be accomplished by dividing the 
semi-circle 15 9, Fig. 52, into a number of equal parts, 
thereby locating a point as 5, which divides the semi- 
circle into two equal parts. From point A of the plan 
draw lines to each of the points of division in one part, 
as shown at 1 A, 2 A, 3 A, 4 A, and 5 A. From point B 
draw lines to each of the points of division in the remain- 



ENDS NOT IN PARALLEL PLANES 87 

ing part as 5 B, 6 B, etc. This, as will be noted, divides 
the whole surface of one half of the object into triangles. 

The elevations of points upon the round end may be 
secured, as is clearly shown by the vertical projectors, 
i.e., lines 11,22,3 3, etc. All lines radiating from 
point A in plan connect points of the semi-circle from 1 
to 5, and all lines radiating from point B connect points 
from 5 to 9, therefore to secure elevations of these lines 
is but a simple matter. 

The True Lengths of Lines. 

It may be here explained that these lines are the plans 
and elevations of lines presumed to be upon the surface 
of the object, and do not represent their true lengths 
except in two instances, i.e., 1 X and 9 Y of the plan. 
Here, as will be noted, 1 A and 9 B of the elevation sup- 
ply those lengths, since lines represented in plan at 1 X 
and 9 Y are parallel to the vertical plane. All others are 
at an angle to the planes of projection, therefore the 
right angled triangle is employed to secure their true 
lengths ; as for example, the distance from point A to 
each of the several points 1, 2, 3, 4 and 5 of the circle, 
also from B to 5, 6, 7 , 8 and 9, represent the length of 
base of a right angled triangle whose hvpothenuse will 
furnish the true length of the line. Therefore we draw- 
in any convenient position lines at right angles to each 
other, as C D and D E, also F G and G H of the diagram 
of triangles, Fig. 52. Set off from points D and G along 
lines D E and G H, distances found in plan, as is clearly 
shown. 

Vertical Hight of Triangles. 

The vertical hight of each triangle is governed by the 
difference in hight of the extremities of the line for which 



88 TRIANGULATION 

the triangle is constructed. Upon examination we find 
that in this example the difference in hight of the ex- 
tremities of those lines can be represented in two dis- 
tances. Since those lines radiating from point A of the 
elevation all terminate in a line which is parallel to / L, 
we use one vertical hight for all triangles employed to 
secure the true lengths of lines radiating from point A 
of the plan or elevation, as shown at C of the diagram 
of triangles. As will be noted, similar conditions prevail 
in the case of all lines radiating from point B, therefore 
the distance from G to F of the diagram of triangles is 
the vertical hight of all triangles employed to secure the 
true lengths of lines radiating from point B. Upon 
drawing lines as shown, i.e., from the points of division 
upon the base lines D E and G H to points C and F, the 
true lengths of these lines are determined. 

The Pattern. 

Having located a number of right lines which may be 
presumed to be upon the surface of the object, and having 
determined their true lengths, we are now in a position 
to place those lines upon the plane of development in 
their correct relative positions. 

Beginning with line 1 X of the plan, we find its true 
length is 1 A of the elevation. Therefore we draw a line 
of this length upon the plane of development, as shown 
at 1 X of the pattern. We note that line 1 A of the plan 
radiates from point 1 and terminates at point A at the 
base of the object. The distance from X to A at the base 
of the object is equal to the distance between points X 
and A of the plan, therefore we may set our compasses 
to a span equal to the distance between X and A. of the 
plan, and placing one point at X of the pattern, describe 
a small arc as at A. 



ENDS NOT IX PARALLEL PLANES 89 

Since the line 1 A radiates from point 1, and its true 
length is as shown in the diagram of triangles, we use 
that length as radius, and with point 1 of the pattern as 
center, to describe a second small arc at A of the pattern, 
thereby locating that point. We note that lines A 1, A 2, 
A 3, A 4 and A 5 all radiate from point A upon the sur- 
face of the object, therefore we use point A of the pattern 
as center, and the true lengths of these lines as radii in 
rotation" to draw small arcs as shown at 2, 3, 4 and 5 of 
the pattern. The upper extremities of said lines must 
lie in these arcs, and at distances from each other equal 
to the distance between points of division of the circles 
shown in plan, since in this example the circle in plan 
is the true form and size of that end of the object. 

Using these spaces, and beginning at point 1 of the 
pattern, the second small arcs are drawn, thereby locating 
points as shown at 2, 3, 4 and 5 of the pattern. Line 5 B 
radiates from point 5 in plan and elevation, and as the 
true distance between points A and B is shown in the 
elevation, which is 16 inches, it is but a simple matter 
to locate point B of the pattern. The remaining work of 
completing the semi-pattern is but a repetition of the 
earlier operations, using the true length of each line in 
rotation, and should hardly need further explanation. 

The following chapter is devoted to a second demon- 
stration wherein identical results are obtained by the 
use of diagrams whose appearance may, at times, lead 
the novice to believe that entirely different methods were 
employed to secure the pattern for a form as shown at 
Fig. 50. 

However, upon devoting some attention to each chap- 
ter, it will be noted that the different appearance is due 
to the changed position assumed for the object. 



CHAPTER XII. 

The Pattern for a Fitting Whose Ends Are Not in 
Parallel Planes. Second Demonstration. 

In Chapter XI, methods were discussed which may be 
employed to secure the pattern for an object as illustrated 
at Fig. 50, when it is presumed to occupy a position as 
shown at C, Fig. 51. In this Chapter methods are dis- 
cussed which may be employed to secure identical results 
when the object is presumed to occupy a position as 
shown at A, Fig. 51. 

It may be well to explain that the two demonstrations 
are for the purpose of illustrating the different posi- 
tions the object may be assumed to occupy as regards 
the planes of projection, and yet secure the same results 
in the finished pattern. It should also be understood that 
these demonstrations are not for the purpose of recom- 
mending either. The pattern cutter must decide which 
he can best comprehend and employ. No doubt additional 
positions will be conceived by those who give the matter 
careful attention. 

The elevation is first drawn as shown at 1 9 B A, Fig. 
53, and consists, as before, of a section of the object, 
although here the base line A B is parallel to I L. 

The base line A B now becomes what may be looked 
upon as the edge elevation of a square surface whose 
length of side in this instance is 16 inches. The line 1 9 
becomes the edge elevation of a circular surface whose 
diameter is 14 inches. Measurements as here given may 
be verified from the scale included in Fig. 53. 

90 



ENDS NOT IN PARALLEL PLANES 



91 




92 TR1ANGULATI0N 

Upon referring to Fig. 52 in Chapter XI, it will be 
noted that these so-called surfaces occupy the same rela- 
tive positions as there shown. Since the object consists 
of two equal but opposite halves, and either may be 
duplicated for the other, we shall only concern ourselves 
with one-half — i.e., that half nearest the eye. 

Since the square end is parallel to the horizontal plane, 
the semi-plan of the base becomes a rectangle whose 
dimensions are 8 x 16 inches, as shown in plan at X A 
B Y. The semi-plan of the top or round end becomes a 
semi-ellipse, since the circular surface referred to above 
is oblique to the horizontal plane. To draw this ellipse, 
we assume an oblique supplementary plane parallel to 
the surface to be represented. 

Thus in any convenient position, draw a line parallel 
to 1 9 of the elevation as G L, Fig. 53. This line be- 
comes the intersecting line between a supplementary 
plane and the vertical plane of projection. Since the 
supplementary plane whose intersecting line is G L is 
parallel to the top or round end of the object, we may 
draw upon it, in a position as indicated by the oblique 
projectors 11,22,33, etc., a circle whose diameter is 
equal to the round end, which in this instance is 14 
inches. 

Divide this circle into two equal parts by a line parallel 
to G L. Divide one half of this circle into an equal num- 
ber of equal parts as shown at 1, 2, 3, 4, etc., of the circle 
upon the oblique plane. Project these points of division 
to line 1 9 of the elevation, thereby locating points whose 
positions have previously been established upon the 
oblique plane. Points thus secured in elevation, with 
the exceptions of 1 and 9, may be looked upon as the end 
elevations of lines which connect portions of the circle 
nearest the eye to points of the right line whose projection 



ENDS NOT IN PARALLEL PLANES 93 

upon the oblique plane is 1 9. Since these lines are repre- 
sented in elevation by points, they must be perpendicular 
to the vertical plane, therefore parallel to the horizontal 
plane of projection. The plans of said lines will be 
found in lines let fall from points 1 , 2, 3, 4, etc., of the 
elevation, perpendicular to I L. 

Importance of a Knowledge of Orthographic 
Projection. 

The question now presents itself, "What is the dis- 
tance from that point to the point beyond?" This is a 
question which frequently arises in pattern development, 
and a correct answer is the solution of many problems. 
However an ability to answer all such questions can only 
be acquired by a knowledge of Orthographic Projection, 
the fundamental principles of which were discussed in 
Chapters IX and X. 

Since we have confined ourselves to developing the 
pattern for one half of the object, we may look upon 
the line X F 9 of the semi-plan, Fig. 53, as a line which 
divides that diagram into two equal parts, and is, of 
course, farthest from the eye. Therefore we are chiefly 
concerned in determining the lengths of lines which con- 
nect these points of the round end nearest the eye, and 
terminate at X Y 9. 

As previously stated, the plans of lines whose end 
elevations are in points 1, 2, 3, 4, etc., of the elevation, 
are found in lines drawn from these points perpendicular 
to line / L, as 2 2, 3 3, 4 4, etc. The intersections of said 
lines with line X Y 9 must be the extremities of those 
lines which are farthest from the eye. Upon determining 
their lengths, their extremities nearest the eye may be 
located. 

The line 1 9 of the profile divides that circle into equal 
parts, therefore the plan of point 1 is a point upon line 



94 TRIANGULATION 

X Y 9 as shown at 1 of the plan. The length of line 2, 
whose end view is at 2 of the elevation, is shown at 2 a of 
the profile, and this distance set off from line X Y 9 gives 
us point 2 of the plan. The length of line 3 in plan is 3 b 
of the profile, and 4 is 4 c of the profile. In like manner 
the lengths of additional lines shown are set off from 
the line X Y 9 of the plan to secure points through which 
the curve is traced. This is a plan of the top of the 
object. 

Points thus located are used as points to which lines 
are drawn from the vertices of angles at A and B, to 
secure the plans of lines presumed to be upon the surface 
of the object, thus dividing said surface into triangles. 
The elevations of said lines are now drawn in the same 
general manner as was explained in the first demonstra- 
tion, and here shown in the elevation, Fig. 53. 

In this example as with the first, the true lengths of 
lines whose plans are X 1 and Y 9 are found in elevation 
in lines 1 A and 9 B. The true length of each remaining 
oblique line will be found in the hypothenuse of a right 
angled triangle whose base is equal in length to the plan 
of the line, and whose perpendicular is equal in length 
to the difference in hight of the extremities of that line 
from / L, as clearly shown in the diagram of triangles. 

Upon examination of the diagram of triangles, Fig. 
53, it will be noted that the lines C D and D E have been 
drawn at right angles to each other, and form two sides 
of the right angled triangle from which we secure the 
true lengths of lines. The hypothenuse of each is located 
by first setting off from D along the line D E, the length 
of the line in plan, and locating points along line C D 
equal in distance from D to the vertical hight of the line 
shown in elevation. 

As for example, the length of line Aim plan is set 



ENDS NOT IN PARALLEL PLANES 95 

off as shown from D on the line D E. The vertical hight 
of line A 1 shown in elevation is at C, and a line is drawn 
to said points as shown at 1 A of the diagram of tri- 
angles. This line, as will be noted, is the true length of 
line A 1 upon the surface of the object. It is only to be 
remembered that the true lengths of all remaining lines 
arc secured in a similar manner, as clearly shown by 
construction lines in the diagram of triangles. 

In transferring the lengths of lines, we may use the 
same general methods as explained in previous demon- 
strations. The first line to be placed upon the plane of 
development is 1 X, whose true length is shown at A 1 
of the elevation. The lines X A, A B, and B Y are in 
their true lengths in plan, and the true length of line 1 A 
is found in the diagram of triangles as above described. 
In fact the length of each line whose true length is not 
shown in plan or elevation is found in the diagram of 
triangles. 

If it is remembered that the true distance between in- 
dicated points of the top is found between similar points 
upon the circle shown upon the supplementary plane, and 
as the diagrams have been drawn to the scale included, 
one should have little difficulty in securing a clear under- 
standing of this, a second demonstration of developing 
the pattern for an object whose specifications were given 
in Chapter XL 



CHAPTER XIII. 

A Transitional Elbow from Round to Rectangular 

1 he fitting illustrated at Fig 54, whose most common 
use is found in furnace work, supplies us with a com- 
paratively simple example in pattern development. A 
fitting f f this class may be modified in many ways and 




Fig. 54. Photographic View of a Transitional Elbow. 

not materially alter the method of securing its pattern. 
For convenience, we shall presume it to be made to given 
dimensions, which may be verified by the use of the scale 
in Fig. 55. 

We propose to secure the pattern for a fitting which 
will make a right angled connection between a round and 

96 



TRANSITIONAL ELBOWS 97 

a rectangular pipe, whose dimensions are as follows: 
Diameter of round pipe 15 inches, cross sectional dimen- 
sions of the rectangular pipe of 6 x 24 inches. Their 
relative positions are shown in the elevation Fig. 55, 
since the line A B of that view is the edge view of the 
lower extremity of the rectangular pipe, and that portion 
included within the lines 1 9,9 a, a b, and b 1 represents 
one extremity of the round pipe. It will be noted that 
the round pipe is here shown to be cut oblique to its axis, 
and to this the irregular portion of the fitting is con- 
nected. 

The Irregular Portion. 

The irregular portion, with which we are chiefly con- 
cerned, becomes a form making a transition from ellip- 
tical to rectangular, the ends of which are not parallel. 
This presents a problem which is not unlike some previ- 
ously explained ; however, if we attempt to follow those 
methods it will be difficult to determine the exact form 
and size of the elliptical end. This can be accomplished 
theoretically, but in practice it is somewhat difficult to 
work with the accuracy demanded when the irregular 
portion is to be seamed to the round collar. Other 
methods than those previously shown will give more ac- 
curate results in practice, and are illustrated and ex- 
plained in this demonstration. 

Plan and Elevation. 

The most simple diagram which will represent the 
object shown at Fig. 54 is a section, and shown as an 
elevation in Fig. 55. At least two views are necessary 
to enable us to secure the true lengths of lines presumed 
to be upon the surface of the object, therefore the next 
work will be to draw the second view, which is a semi- 



98 TRIANGULATION 

plan. Since the object is composed of two equal but 
opposite halves, it is only necessary to represent one-half 
in plan. 

To secure this semi-plan, we let fall perpendicular lines 
from points A and B of the elevation, and make them 12 
inches in length from / L, as shown at X A and Y B. 
Upon drawing the line A B as shown, the semi-plan of 
the rectangular pipe in its assumed position is completed.. 

That portion of the elevation included within the lines 
91,1 b,b a, and a 9 represents a round collar which has 
been cut obliquely, and to secure a plan of this it will be 
most convenient to locate a number of elements upon 
its surface. Therefore in any suitable position to the 
right of line G L, draw a semi-circle whose diameter is 15 
inches. This constitutes a profile, or an end view of the 
round collar upon a profile plane, of which G L is the 
intersecting line between it and the primitive vertical 
plane. 

Divide the semi-circle into a number of equal parts as 
shown. From said points of division right lines are 
drawn parallel to line I L to intersect the miter line 1 9. 
These points of intersection upon the miter line, as at 
1, 2, 3, 4, etc., are now looked upon as the end elevations 
of lines which are perpendicular to the vertical plane of 
projection ; therefore their plans will be found in lines let 
fall from said points at right angles to / L, as shown. 
The lengths of these lines are found in the distances 
similarly numbered points of the semi-circle are from the 
line G L. These lengths are set ofif from the line I L in 
plan, as also shown. This, as will be noted, locates 
points through which the curved line is traced to com- 
plete a semi-plan of the oblique end of the round collar. 

Points located as above described may now be con- 
nected by lines to points A and B as shown, to supply 



TRANSITIONAL ELBOW 



99 




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100 TRIANGULATION 

the plans of lines presumed to be upon the surface of the 
object, and utilized as measuring lines to be transferred 
to the plane of development, although their true lengths 
must yet be determined. The elevations of lines whose 
plans are shown at A 1, A 2, A 3, A 4, A 5, B 5, B 6, 
B 7 ,B 8 and B 9 of the semi-plan, Fig. 55, are secured by 
drawing lines from points 2 to 5 inclusive to A, and from 
5 to 8 inclusive to B, as shown in the elevation. 

True Lengths of Lines. 

The true lengths of these lines are secured by the use 
of the right angled triangle, as shown in the diagram of 
triangles, where the line D E is used as the perpendicular 
for all, with a common vertex at D. The horizontal lines 
of the elevation, as 1 1,2 2,3 3, etc., represent the hight 
of the lower extremities of these lines — i. e., the lower 
extremity of line A 2 is at point 2 of the elevation, and so 
on for all lines shown. 

The vertical distance from those lines to line B D is the 
perpendicular hight of each triangle, and the lengths of 
these lines in plan is the base of each triangle, as will be 
clearly shown if the reader, by the use of his compasses, 
will compare measurements. 

Having determined the true lengths of lines shown in 
plan and elevation, we may proceed to transfer these 
lengths to the plane of development in the process of 
securing the pattern. 

To Develop the Pattern. 

There is shown in plan at^Ila triangular surface 
whose edge elevation is the line A 1 of that view. Since 
the line X 1 is parallel to I L, its true length is A 1 of the 
elevation, and this length is transferred to line X 1 of the 



TRANSITIONAL ELBOW 101 

semi-pattern. From point X of the semi-pattern draw a 
line perpendicular to X 1, and set off a distance from X 
equal to the length of line A X of the semi-plan, as shown 
at A of the semi-pattern. Upon drawing the line A 1, 
the true form of the triangular surface has heen placed 
upon the plane of development. 

It may be well to here explain that where right tri- 
angular surfaces are shown in plan, those triangles will 
be right angled in their true form; therefore we may 
transfer them to the plane of development in the manner 
as above explained, or employ our compasses to transfer 
the length of each of the lines of which the triangle is 
composed. The author has made a practice of employ- 
ing his compasses to transfer the lengths of lines. If the 
resulting triangle is right angled, a portion of his work 
is proven. 

Upon examination we find five lines radiating from 
point A of the plan or elevation, Fig. 55, whose true 
lengths are shown in the diagram of triangles. There- 
fore we may use these lengths as radii, and with point A 
of the pattern as center to describe small arcs as shown 
at 2, 3, 4 and 5 of the semi-pattern of the irregular por- 
tion. Since the distances between the lower extremities 
of these lines are not shown in plan and elevation, they 
must be determined. As has been previously stated, this 
can be done approximately by securing the true form of 
the oblique section of the round pipe, and will be ex- 
plained below. For the present we shall pursue the more 
accurate and simple method of first developing the pat- 
tern for the round collar. As this may be looked upon 
as the pattern for one section of an elbow in round pipe, 
the miter line of which has been located, it is not an ex- 
ample in triangulation. 

The same elements are employed to secure this pattern 



102 



TRT ANGULATION 



as are shown in elevation, therefore the lower extremities 
of lines whose true lengths are shown in the diagram of 
triangles must intersect these elements of the cylinder 
at the miter cut. This being understood, the reader will 
realize that the distance between lines as A 1, A 2, A 3, 
etc., at the lower extremity of the irregular portion, must 
be the same as shown between similarly designated ele- 
ments upon the miter cut of the round collar. Therefore 




Fig. 56. Diagram Employed to Secure the True Form of the 
Oblique Section of a Cylinder. 

we use these distances in rotation, beginning at / to 
describe additional small arcs, as shown at 2, 3, 4 and 5 
of the pattern for the irregular portion. Having located 
points A and 5, or line A 5 of the pattern, we may add 
the triangular surface shown in plan and elevation at 
A 5 B. 

The line A B is in its true length in either plan or 



TRANSITIONAL ELBOW 103 

elevation ; therefore we may set our compasses to a span 
equal to the length of line A B of the elevation, and 
placing one point at A of the pattern, describe the small 
arc shown at B. From the diagram of triangles we secure 
the true length of line 5 B, which we use as radius, with 
point 5 of the pattern as center, to draw the second small 
arc as also shown at B. 

There are five lines radiating from point B in eleva- 
tion, whose positions may be located upon the plane of 
development in the same general manner as was ex- 
plained for those radiating from point A. Presuming the 
extremities of line B 9 to have been located, the remain- 
ing triangular surface, B Y 9 may be added, since from 
the plan we secure the true length of line B Y, and in the 
elevation the true length of line Y 9 is found in line B 9. 

It is by no means necessary that the semi-circle or 
profile shall be divided into the number of parts shown 
in this demonstration. Divide the profile into any equal 
number of parts desired. More parts will increase the 
work, and increase the accuracy to some extent. 

Form of the Round Collar at the Mitered End. 

The true form of the oblique section of a cylinder is 
an ellipse,* the minor diameter of which is equal to the 
diameter of the cylinder. The major diameter is de- 
pendent upon the length of the miter line, when it can 
be looked upon as the edge view of a plane which has cut 
said cylinder. Therefore, in this instance, the true form 
of the round collar at the miter cut is an ellipse whose 



* Xo part of a true ellipse is a part of a circle, therefore any method 
which involves arcs drawn from centers will not produce a true ellipse. 
Approximate ellipses drawn in this manner are sometimes known as false 
ellipses, and in some instances are found to be sufficiently accurate. 



104 TRIANGULATION 

minor diameter is 15 inches, and whose major diameter 
is approximately 18 inches, and can be drawn by any 
method which secures a true ellipse. 

Perhaps the most desirable course to pursue is to se- 
cure this ellipse by projection as follows : Draw the 
side elevation of the cylinder as shown in Fig. 56. Draw 
a semi-circle whose diameter is equal to that of the 
cylinder as shown in the profile. Draw a line as A B, 
parallel to the miter line. 

We now have what may be looked upon as the side 
elevation of one half of the cylinder, showing the miter 
line, and drawn upon the primitive vertical plane. To 
the right of the elevation is the profile plane, with the 
line G L as the intersecting line between this and the 
primitive vertical plane. To the right and below, the true 
form is shown upon an oblique supplementary plane, 
which is also perpendicular to the primitive vertical 
plane. This is a projection of that portion of the cylinder 
represented in elevation by line 1 9 upon a plane parallel 
to it, when viewed as indicated by arrows. 

The semi-circle shown as a profile is divided into a 
number of equal parts, and from these points of division 
lines are drawn parallel to I L to intersect the miter line. 
From points thus secured as 1, 2, 3, etc., upon the miter 
line, project lines perpendicular to A B as shown. From 
the intersections of these lines with A B, set off distances 
as found from line G L to similarly numbered points of 
the semi-circle, thereby securing points through which 
the curve of the ellipse may be traced. 

The distances between points of the ellipse are sub- 
stantially the same as found by the more simple method 
of first determining said distances by developing the pat- 
tern for the round collar. We can, if we choose, use that 



TRANSITIONAL ELBOW 105 

form for the base, and develop the pattern in the same 
general manner as was explained in Chapter XI. How- 
ever, this process will be found to be more complicated, 
and less accurate in pattern problems of this class. 



CHAPTER XIV. 

Transitional Offset from Round to Rectangular. 

Throughout this work the author has endeavored to 
direct the reader's attention to the importance of an 
understanding of the principles involved. To those who 
have followed the work, it must have become evident that 
the solution of all problems coming under the head of 




Fig. 57. Photographic View of a Transitional Offset. 

Triangulation presents a great sameness. An under- 
standing of how best to draw the first diagrams to repre- 
sent the object for which a pattern is required is an im- 
portant factor. 

Attention is here directed to the transitional offset as 
shown pictorially at Fig. 57. The offset is for the pur- 

106 



TRANSITIONAL OFFSET 107 

pose of making connection between a round and a rect- 
angular pipe. The diagrams shown have been drawn 
to the scale appended. The diameter of the round pipe is 
14 inches, with cross-sectional dimensions of the rect- 
angular pipe of 7 x 20 inches, whose relative positions 
are shown in plan and elevation, Fig. 58. 

The Capacity of the Fitting. 

It will be noted upon examination of the elevation that 
both the round and rectangular pipes have been cut ob- 
liquely. This has been done, as has been previously ex- 
plained, for the purpose of preserving the capacity of 
the fitting, whose most common application is found in 
furnace work, although similar examples will occa- 
sionally come before the sheet metal worker in other 
lines. 

The specification tells us that in this instance, a trans- 
ition is required between a 14 inch round pipe and a 
7 x 20 inch rectangular pipe, with an ofTset of 8 inches as 
shown, and to be accomplished in a distance of 14 inches. 
With the above information in hand, it is but a simple 
matter to draw an elevation as shown by the boundary 
lines of that diagram, Fig. 58. 

An Analysis of the Fitting. 

The positions of lines which represent the connecting, 
or miter lines between the collars and the center irregular 
portion are by no means arbitrary, although their loca- 
tions must be governed to some extent by existing con- 
ditions. Having located these lines to our satisfaction, 
somewhat as shown in elevation, we find from an analysis 
of the problem that the fitting is composed of three parts, 
i.e., we have a round collar with one end cut obliquely, 
which may be looked upon as one piece of an elbow in 



108 



TRIANGULATION 




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TRANSITIONAL OFFSET 109 

round pipe. A collar for the rectangular pipe, which is 
also cut obliquely, and a center portion, which is an 
irregular one, rgaking a transition from elliptical to rect- 
angular, the ends of which are not parallel. To develop 
the pattern for this piece is the subject matter of this 
chapter. 

It may be explained that to secure this pattern it is by 
no means necessary to follow the course here recom- 
mended, although some course must be pursued which 
will enable the worker to determine the true lengths of 
lines presumed to be upon the surface of the object. Since 
the diagrams shown in Fig. 58 are about as simple as the 
nature of the problem will permit, one will hardly go 
astray if he follows them absolutely. 

The Plan. 

After having drawn an elevation, which is in reality a 
section of the object, the next step is to secure a plan, or 
at least a semi-plan, as the fitting is here shown to be com- 
posed of two equal but opposite halves. 

The semi-plan of the round collar is a semi-circle with 
a diameter of 14 inches, and in a position as shown. A 
semi-plan of the rectangular collar in its assumed position 
is a rectangular diagram, as X A B Y of the plan, and in 
position as clearly shown by the vertical projectors. 

Divide the semi-circle of the plan into an equal number 
of equal parts, thereby locating" one point which divides 
it into two equal parts as 5 of the plan. Draw lines from 
all points thus located upon the arc 1 5 to point A, and 
from points upon the arc 5 9 to point B as shown in plan. 
This secures the plans of lines which we presume to be 
upon the surface of the fitting. The elevations of the 
above lines are secured by projecting lines from points 
as 1, 2, 3, 4, etc., of the semi-circle, perpendicular to / L 



110 TRIANGULATION 

to intersect the oblique line 1 9 of the elevation as at 
1, 2, 3, 4, etc., of that view. From these points upon the 
oblique line 1 9, lines are drawn to points A and B as 
shown in elevation to secure elevations of lines whose 
plans have previously been drawn. 

True Lengths of Lines. 

Having now before us the plans and elevations of lines 
which are presumed to be upon the surface of the fitting, 
it remains to determine their true lengths and relative 
positions, also to place them upon the plane of develop- 
ment in those lengths and positions. The plan of the 
line supplies the base, and the difference in hight of the 
extremities of the line as shown in elevation, supplies 
the perpendicular of a right angled triangle, whose 
hypothenuse is the true length of the line, as is clearly 
shown in the diagram of triangles. 

As for example, we select line 3 A of the plan; its 
elevation is 3 A of that view. The difference in hight of 
the extremities of that line has been determined by draw- 
ing lines to the right from points A and 3 of the eleva- 
tion, and parallel to I L as A C and 3 f. Then C / is the 
difference in hight of the extremities of line 3 A, or the 
perpendicular of a right angled triangle whose base is 
equal in length to line 3 A of the plan, or / g of the dia- 
gram of triangles. Similar work and reasoning will 
enable one to secure the true lengths of all lines radiating 
from points A and B of the plan. When the true lengths 
of all lines have been secured as shown in the diagram 
of triangles, the pattern can quickly be developed as 
shown. 

The Pattern. 

Beginning with the line whose plan is 1 X, we find 
its true length in line 1 A of the elevation. This length 



TRANSITIONAL OFFSET 111 

is set off upon the plane of development as at X 1 of the 
pattern. The true length of line 1 A of the plan is found 
in 1 A of the diagram of triangles. Its lower extremity 
is at point 1 in all views, and its upper extremity at A 
is at a distance from X equal to the length of line X A 
of the plan. 

We have in this demonstration four additional lines 
radiating from point A, whose true lengths are shown in 
the diagram of triangles, and since said lines radiate 
from a single point, we have only to determine the dis- 
tances between their lower extremities. Should the reader 
experience any difficulty in comprehending this, he is 
advised to refer to Chapter XIII, where this feature was 
explained to some length. 

Upon developing the semi-pattern for the round collar 
as shown, these distances are secured and used as radii to 
draw small arcs in rotation, thereby locating the lower 
extremities of lines as shown at A 1, A 2, A 3, A 4 and 
A 5. Presuming that the line A 5 has been located upon 
the pattern as shown, an examination of the plan and 
elevation shows that the triangular surface A 5 B should 
now be added. 

The true length of line B 5 is found in the diagram of 
triangles. Using this length as radius and with point 5 
of the semi-pattern as center, a small arc is drawn as 
shown at B of the pattern. The true distance from A to 
B is shown in the length of line A B of the elevation. 
Therefore that length is used as radius with point A of 
the semi-pattern as center, to draw the second small arc 
at B, thereby locating the upper extremity of line B 5 
upon the pattern as shown. 

There are also in this example, four additional lines 
radiating from point B. As before, the true lengths of 
these lines are found in the diagram of triangles, which 



112 TRIANGULATION 

may be used as radii in rotation, using point B of the pat- 
tern as center, to describe small arcs as shown at 6, 7, 8, 
and 9. The true distances between these points are found, 
as before, between similarly numbered elements of the 
round collar upon the miter cut. Using these in rotation, 
the second small arcs are drawn to intersect the first, 
thereby locating points which are in reality the lower 
extremities of lines B 6, B 7 ,B 8 and B 9, as shown upon 
the pattern. Having drawn the line B 9 upon the semi- 
pattern, the remaining triangular surface as there shown 
is located by first locating point Y. 

We find upon examination that the true distance be- 
tween points B and Y is the length of B Y of the plan, 
and that the true distance between points 9 and Y is the 
length of line 9 B of the elevation, thereby enabling us, 
by the use of our compasses, to locate point Y as shown 
upon the pattern. 

Upon drawing lines B Y and Y 9 the semi-pattern is 
complete, which when duplicated, and formed in the 
opposite direction, will combine with the pattern here 
shown, to complete the irregular form necessary to make 
connection between the round and rectangular 
when cut obliquely as shown in elevation. 



CHAPTER XV. 

A Three Pieced Tapering Elbow. 

The solution of the problem here presented should in- 
terest the pattern cutter, although the demand for a 
fitting of this class is limited. An endeavor is made in 
this example to satisfy a popular demand for something 
out of the ordinary. Therefore the fitting, as shown in 
Fig 59, has been presumed to be what is commonly known 




Fig. 59. Photographic J lew of Three-pieced Tapering Elbow. 

as flat on one side. This necessitates the developing of 
the pattern for the whole irregular portion, as one half 
cannot be duplicated for the other. 

In examples of this nature there are no additional 
principles to be applied, but it adds to the complication 
of lines shown in the diagrams. Unless they are given 
careful attention, no doubt these diagrams will appear 

113 



114 



TRIANGULATION 



complicated. On the other hand, some attention to this 
will pave the way for one to successfully develop patterns 
for those objects which have more or less of a distorted 
form, since some principles as here explained, may be 
applied to practically all such examples. 

The scenographic representation of the object, and the 
planes within which it is presumed to be situated, as 
shown at Fig". 60, has been introduced in an endeavor to 
show to some extent, in a pictorial way, the value and 




Fig. 60. Scenographic Reproduction of Elbow and Lines 
Presumed to Be Upon Its Surface, Also the Planes Within 
Which It Is Presumed to Be Situated. 

positions of lines whose plans and elevations are shown 
in Fig. 61. 

It will be noted that the fitting consists of three parts, 
i.e., there are two round collars placed in positions which 
are in this instance, at right angles to each other, and 
one piece forming a center portion. This demonstration 
deals chiefly with the center portion, since the round 
collars are but parts of elbows in round pipe of different 
diameters. The center portion is in reality a transition, 



TAPERING ELBOW 115 

or a change of form to make the necessary connection as 
shown. The ends of this piece are elliptical, since said 
ends connect cylinders which have been cut obliquely. 
Therefore we could determine the size and form of the 
oblique ends of the round pipes or collars, and consider the 
center portion as a transition whose ends are elliptical 
and not in parallel planes, although perhaps the more 
simple course to pursue is as will be here explained. 

The Elevation. 

The most simple diagram to be drawn which will repre- 
sent a fitting as illustrated at Fig. 59, is an elevation 
somewhat as shown at Fig. 61. The author has used the 
word "somewhat" for the reason that considerable 
change may be introduced into the fitting and yet employ 
the same methods of securing its patterns. The positions 
of the miter lines are by no means arbitrary, although in 
this instance they have been given the same inclination 
that would prevail in a three pieced elbow in round pipe.* 

The Plan. 

Having drawn the diagram to represent the object, 
which is to some extent a section as shown by the bound- 
ary line of the elevation, Fig. 61, the plan may be pro- 
ceeded with. The large circle in plan is drawn to the 
diameter of the large collar, and in a position as shown. 
To secure a plan of the small collar involves somewhat 
more detail. The small collar in this example is parallel 
to the horizontal plane, with one end cut obliquely. A 
plan of the oblique end becomes an ellipse. To secure this 
ellipse, we presume lines to be upon the surface of the 



* If the reader desires information on the development of patterns for 
elbows in round pipe, he will find that the book "Elbow Patterns for all 
forms of Pipe" explains this in every detail. 



116 



TRIANGULATION 



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TAPERING MI. BOW 117 

collar, and to locate those lines we introduce the profile 
plane. As for example, a circle is drawn to the right of 
the elevation whose diameter is equal to the diameter of 
the small collar, and in a position as shown. For con- 
venience, this circle is divided into an equal number of 
equal parts, and from said points of division, lines are 
drawn parallel to line I L to intersect the miter line of 
the small collar. Such lines are known as elements of 
the cylindrical surface, and their left hand extremities 
are now points in the ellipse. In other words, those lines 
terminate at the miter line, since said miter line is looked 
upon as the edge view of a plane which has cut said 
cylinder, and since this plane is represented in elevation 
by a right line, it must be perpendicular to the primitive 
plane of projection. 

From points secured by the intersections of the ele- 
ments of the cylinder with the miter line, perpendicular 
lines are drawn of indefinite lengths below the line / L, 
then points in the ellipse must lie in some points along 
these lines. From what has been explained in previous 
demonstrations, it should be understood that when the 
profile of the small collar was drawn, an additional plane 
was presumed which is known as a profile plane, and the 
line A B, presuming it lies in the plane of the paper, may 
be looked upon as a vertical axis upon which said profile 
plane may be presumed to revolve. 

This is shown in a pictorial way at Fig. 60, where L M 
O N represents the vertical plane, X P O the hori- 
zontal plane, and M R Q the profile plane. The line 
N is the intersecting line between the vertical and hori- 
zontal planes, while M corresponds to the line A B of 
the elevation, Fig. 61 . Since the line A B of the elevation 
lies in the vertical plane, its plan must be a point in line 
/ L, as at B, Fig. 61. Tf perpendicular lines be drawn 



118 TRIANGULATION 

from the points of division of the circle in elevation to 
intersect line / L, said lines must be distant from line 
A B equal to the distance points are from which these 
lines have been drawn from the primitive vertical plane. 

If these distances are revolved about the point B upon 
the horizontal plane, then said distances are located upon 
that plane. Lines now drawn through these points so 
revolved, and parallel to the line / L, will intersect verti- 
cal lines drawn from the miter line in points which are 
located in the ellipse as at numbered points of that 
diagram. 

Relative Position of the Large Circle in Plan. 

It may be explained that the large circle in plan which 
represents the large collar must be placed in its correct 
relative position. For example, if the fitting is to be made 
in two equal halves, then the axis of each collar will be 
represented in plan at the same distance from line / L, 
but if in an instance as here shown, where one side of the 
fitting is flat and composed of two unequal parts, then 
the circle in plan which represents the large collar must 
be so placed as to allow one side of the fitting to be repre- 
sented in plan by a line parallel to I L. 

Presuming lines to have been drawn whose intersec- 
tions upon the horizontal plane are points in the ellipse, 
as shown in plan, Fig. 61, we are now in a position to 
locate lines which are looked upon as being upon the 
surface of the object, somewhat as follows : Divide the 
large circle in plan into the same number of equal parts 
as the small circle of the profile has been divided into. 
Draw lines from each point of the large circle to a 
similarly numbered point of the ellipse, as 1 1, 2 2, 3 3, 
etc. This secures plans of the above spoken of lines. To 
secure the elevations of said lines, project vertical lines 



TAPERING KI.BOW 119 

from the points of division of the large circle to intersect 
the miter line of the large collar in elevation. Lines 
drawn, as shown by full lines upon the irregular portion 
in elevation, supplies elevations of those lines whose plans 
are 11,22,33, etc. 

A Practical Demonstration. 

If the reader has any difficulty in comprehending this, 
he is advised to lay out two collars something like those 
whose semi-patterns are shown in Fig. 61, with a number 
of equi-distant parallel lines, as shown at 1, 2, 3, 4, etc., 
of those patterns. Form them, and secure them in posi- 
tions as suggested by the diagrams. He may then pre- 
sume to draw strings between the extremities of the equi- 
distant lines, using care that the first one is from the 
longest line of each collar. He will then find that the 
strings so drawn will include the form for which the 
pattern is required, and the strings may be looked upon 
as elements of that surface. 

The reader may draw upon his imagination to see this 
in Fig. 60, where lines as a a, b b, c c and d d are the 
equi-distant parallel lines, or elements of the large collar. 
Lines as y y, x x, and w w are the equi-distant parallel 
lines or elements of the small collar, and lines as a y, b x, 
and c w represent the strings. As these strings include 
the surface of the required form, it is evident that we 
have only to determine the length of each string and the 
distance they are from each other to develop the pattern. 

However, since the strings represented by the full lines 
will not divide the surface into triangles, we are obliged 
to introduce additional lengths of string, as shown by 
the broken lines. These must be represented in plan to 
secure the pattern from the diagrams. This is ac- 
complished, as will be noted, when the broken lines are 
drawn as 1 2, 2 3, 3 4, etc., in Fig. 61, 



120 triangulation 

The True Lengths of Lines. 

With the plan and elevation complete as shown at Fig. 
61, we now proceed to secure the true lengths of lines pre- 
sumed to be upon the surface of the object, i.e., we may 
construct our triangles. This is but a simple operation 
if it is remembered that the plan of the line supplies the 
base, and from the elevation the perpendicular is secured. 
For example, we may draw indefinite horizontal lines 
through the points of division of the profile as shown. 
From the intersections of lines presumed to be upon the 
large collar with its miter line as at a, b, c, d, additional 
horizontal lines are also drawn. 

In any convenient position we may draw a perpen- 
dicular line as C D of the diagram of triangles, and pre- 
sume the perpendicular of a number of triangles to lie 
in this line. For example, we select the line 1 1 of the 
plan and set off its length from C D upon the horizontal 
line drawn from a as shown. The point C represents the 
vertical hight of the upper extremity of that line, there- 
fore upon drawing a line as 1 1 of the diagram of tri- 
angles, the true length of that line is before us. 

This operation must be repeated for each line repre- 
sented, since there is no guarantee that any two will be 
of the same length. It should also be remembered that 
the true lengths of those lines, as 12,23,3 4, etc., shown 
as broken lines, must also be secured. This is ac- 
complished in the same general manner as has been ex- 
plained for the full lines, and shown in the diagram of 
triangles where the upper extremities terminate at line 
E F. 

The Pattern. 

To locate lines upon the plane of development which 
we have presumed to be upon the surface of the object, 



TAL'KRING ELBOW 



121 



in their correct lengths and positions, now becomes a 
simple, although a somewhat prolonged operation. In 
practice it may be well to develop the pattern as the true 
lengths are secured. This course will very likely render 
one less liable to error, inasmuch as each length may be 
utilized when determined, thereby avoiding to some 
extent, that complication of lines shown in the diagram 
of triangles. 




Fig. 62. Pattern for Center Piece of Elbow. 



The distance lines are from each other at their ex- 
tremities is clearly shown upon the mitered ends of the 
collar patterns, which we shall presume to have been 
first developed. Since these patterns are but portions 
of elbows in round pipe, we will pass directly to the pat- 
tern for the center piece. In examples of this nature it 
may be well to first place the longest full line as 1 1 upon 
the plane of development in its true length, as found in 
the diagram of triangles. By working each way from 
this line so located, we may avoid some additional oppor- 
tunity for error. 

No attempt will be made here to describe in detail the 
method of locating each and every line shown, since that 
simply involves duplicate operations, as has been fre- 



122 TRIANGULATION 

quently explained in foregoing demonstrations. How- 
ever, we will select a few lines as an example, and if the 
reader comprehends the methods of locating these, he will 
have little difficulty in completing the pattern as shown. 
For example, first draw line 1 1 in its true length upon 
the plane of development. The lower extremities of line 
2 2 and 16 16 are distant from the lower extremity of line 
1 1 equal to the distance between elements 1 and 2 at the 
miter cut of the semi-pattern for the large collar as in o. 
Fig. 61. Therefore we set our compasses to a span equal 
to the distance between points m and o, place one point 
:at the lower extremity of line 1 1 of the pattern Fig. 62, 
and describe small arcs as shown at id and 2 at the base. 
It may be explained that for convenience in this example, 
that end of the center piece which joins the small collar 
has been designated as the top, and that portion which 
joins the large collar as the base. 

The upper extremities of line 2 2 and 16 16 are distant 
from the upper extremity of line 1 1 equal to the distance 
between points n and p of the semi-pattern for the small 
collar, Fig. 61. Therefore we set our compasses to the 
distance n p of the pattern for the small collar, place one 
point at the upper extremity of line 1 1 of the pattern, 
Fig. 62, and describe small arcs as shown 2 and 16 at the 
top. The extremities of lines 2 2 and 16 16 must now lie 
in some points of these arcs. 

To determine the exact location of the above spoken 
of points the broken lines are employed. In other words, 
if we can determine the true lengths of lines 1 2 and 1 16 
we can definitely locate points 2 and 16 at the top of the 
pattern. Therefore we set our compasses to a span equal 
to the length of line 1 2 found in the diagram of triangles, 
and placing one point at 1 of the base, describe the second 
small arc as at 2 of the top. With compasses set to a 



TAPERING ELBOW 123 

span equal to the true length of line 2 2 found in the 
diagram of triangles, place one point at 2 of the top of the 
pattern, and describe the second small arc as shown at 2 
of the base. With compasses set to a span equal to the 
true length of line 16 16, also found in the diagram of 
triangles, place one point at 16 of the top and describe 
the second small arc shown at 16 of the base. Lines may 
now be drawn as shown at Fig. 62 to complete what may 
be looked upon as two sections of the pattern for the 
center piece. To complete the pattern, these operations 
just described are continued, using each true length found 
in the diagram of triangles, as is clearly shown by the 
construction lines. 

It will be noted that the broken lines shown in plan 
upon that portion of the object which lies furthest from 
the eye, connect points in a reverse order from those 
shown nearest the eye. This not only allows one line in 
elevation to represent two lines in reality, but allows us to 
work both ways from line 1 1 of the pattern. 

When the pattern is completed it must be formed in 
the proper direction to allow it to be placed in position as 
shown in plan. Care should also be used in connecting 
the collars, i.e., lines as 1 1, 2 2, etc., should be continuous, 
or the fitting will be distorted, 



CHAPTER XVI. 

The Ship's Ventilator. 

The ship's ventilator as illustrated at Fig. 63, should be 
of interest to the prospective pattern cutter, chiefly for 
the reason that it suggests principles and methods which 




Fig. 63. Photographic View of a Ship's Ventilator with a Round 
Mouth. 

may be introduced in the development of patterns for 
many fittings of a similar nature. 

To the reader who has followed this work it should 
have become evident that the form of the ends of the ob- 
ject for which a pattern is required must first be establish- 
ed. The author was at one time asked : "How shall I 

124 



SHIP'S VF.NTII.ATOk 125 

proceed to secure the pattern for a ship's ventilator?" 
The reply was : "First determine what form it shall be at 
the miters ; beyond this your patterns are but simple ex- 
amples in triangulation." 

This is a point which is often overlooked by those who 
are slow to realize that in practically all examples where 
triangulation is to be applied the form of the ends of 
the object must first be established. This is precisely 
what is accomplished when a formula is introduced for 
the construction of the diagrams to represent a ship's 
ventilator. When the diameter of base, or pipe to which 
it is to be connected, is the known quantity, the formula 
which follows has been used to some extent : 

Formula for a Ship's Ventilator With a Round 

Mouth. 

Diameter of base X2 = diameter of mouth. 

Diameter of base X l l / 2 = radius of back. 

Diameter of base X ^4 = radius of throat. 

Angle of mouth to the horizontal 80 degrees. 

The form of all pieces to be round at each end, and 
of diameters equal to the lengths of miter lines shown 
in the resulting elevation. 

This formula has been worked out in Fig. 64 to the 
scale appended, presuming the base to have a diameter of 
16 inches, and the fitting to be made in six pieces. It will 
be noted that the back and throat have been divided into 
the same number of parts, i.e., into as many parts as the 
fitting is to have pieces. Lines drawn between these 
points of division represent the miter lines. As has been 
previously explained, each miter line may now be looked 
upon as the edge elevation of a circle whose diameter is 
equal to the length of the line. There must be as many 



126 



TRIANGULATION 



patterns as pieces in the ventilator, although one-half of 
each piece may be duplicated for the other half. 



To Reduce the Problem to Its Simplest Form. 

The most desirable course to pursue in examples of 
this class is to construct separate elevations of each piece, 
with one end parallel to the intersecting line (/ L) as 
shown at Fig. 65. 




.Scale 



80 degrees. 

Radius of 
Throat 4 in. 

Jt L 



Tn''Ti , J'i'r'T l T l T'T' , i:'T l T'T'T l T'T'T'T'T'T'T' , r 

3 6 9 lFt. 15 18 £1 £ 



Fig. 64. Side Elevation of a Ship's Ventilator. 

The elevation in Fig. 65, as will be noted, is an eleva- 
tion of that portion of the object marked A in Fig. 64, it 
having been revolved in such a manner as to place the line 
a b of Fig. 64 parallel to the intersecting line. To trans- 
fer that diagram is but a simple matter if we draw, or 
presume to draw, the line a c Fig. 64, which then divides 



SHIPS VENTILATOR 127 

that section into triangles, and may be transferred by the 
use of our compasses and straight edge. 

Having drawn the elevation of section A, Fig. 64, as 
shown at Fig. 65, the plan is secured by drawing semi- 
circles as shown, i.e., we draw a semi-circle in plan whose 
diameter is equal to the length of the base line, for a 
plan of the base. To secure a plan of the top, a supple- 
mentary oblique plane is assumed whose intersecting line 
in this instance is the oblique line 9 1 of the elevation. 
Above this line, as shown, a semi-circle is drawn whose 
diameter is equal to the length of line 1 9. 

Divide each semi-circle into the same number of equal 
parts as shown. Draw lines from the points of division of 
the semi-circle representing the true form of the top, per- 
pendicular to the oblique line 9 1 to intersect that line as 
shown in points 1, 2, 3, 4, etc. From these points, i.e., 1, 2, 
3, 4, etc., upon the oblique line 9 1 of the elevation, draw 
indefinite lines perpendicular to the line I L. Set off dis- 
tances upon these lines below the line / L equal to the 
length of similarly numbered lines which cross the semi- 
circle representing the true form of the top. A line traced 
through points thus secured supplies a plan of the top, 
together with a number of points which are utilized in 
developing the pattern. 

Draw full lines between points of the same number in 
plan as shown. This supplies plans of lines presumed to 
be upon the surface of the object. However, since these 
lines do not divide the surface into triangles, additional 
lines must be assumed. Plans of these are secured when 
the broken lines are drawn, as 1 2, 2 3, 3 4, etc. 

Tf the reader has any difficulty in comprehending this, 
he may cut from sheet metal or cardboard a form as indi- 
cated by the semi-circle in plan, the elevation, and the 
true form of the top, Fig. 65. When this is cut, it may 



128 



TRIANGULATION 



be bent at an angle of 90 degrees upon the lines which 
represent the base and top in elevation. He will then 
have a form which is illustrated to some extent in Fig. 66, 
although it is there represented as a solid. By placing 
this in the angle formed by the two planes which are at 
right angles to each other, also shown in Fig-. 66, and re- 
membering that the plan of a point is always directly 




Fig. 65. Plan, Elevation, Diagram of Triangles and Semi- 
pattern for One Piece of a Ship's Ventilator. 

beneath it, and that the elevation of a point is always 
back of it, the reader will have before him an illustration 
which should enable him to fully comprehend the plan 
and elevation shown in Fig. 65. 

Fig. 66 also shows, in a pictorial way, both the full 
and broken lines presumed to be upon the surface of the 
object. However, it should be remembered that Fig. 66 



SHIP'S VENTILATOR 129 

is purely pictorial, and will not supply the true lengths 
of those lines. These true lengths must be secured, as in 
all other examples of triangulation, i.e., by the use of the 
right angled triangle. 

The Pattern by Mechanical Methods. 
We sometimes hear of instances where the worker has 
formed the sheet metal or cardboard as suggested above, 
and then wrapped paper about its face to secure the pat- 
tern. This mode of procedure will, of course, supply the 
pattern since our patterns may always be looked upon as 
the envelopment of a solid whose form is that of the re- 
quired object. 

The True Length of Lines. 
Having before us the plan and elevation as shown at 
Fio- 65 we must determine the true lengths ot lines 
whose positions are indicated in that diagram. To ac- 
complish this we may draw indefinite horizontal lines, 
i e parallel to the line / L, from points along the oblique 
line 9 1 of the elevation, and in a convenient position erect 
a perpendicular line as A B of the diagram of triangles 
This determines the lengths of the perpendiculars of all 
triangles. The lengths of lines as found in plan are set 
off from B along the line B L, as shown. As for ex- 
ample the length of line 3 3 of the plan is set off from b, 
thereby locating a point as at a. A glance at the eleva- 
tion shows us that the upper extremity of line 3 3 is at a 
distance above the horizontal plane equal to the length of 
line B b, therefore upon drawing a line as a b, the true 
length of line 3 3 is determined. 

It should be understood that methods as above ex- 
plained must be applied to every line whose plan is not 
parallel to the line / U The true lengths of those lines 



130 



TRIANGULATION 



whose plans lie in or are parallel to the line / L, as 1 1 
and 9 9 are found in the elevation, since a triangle has 
been constructed there whose base is equal to the line 
in plan, as shown in one instance at 1 1 1. Therefore 
it becomes unnecessary to prolong the search for these 
lengths. 

The lengths of broken lines are determined in precisely 
the same manner, although to avoid confusion in this 




Fig. 66. Sccnographic Representation of One-half of One 
Piece of a Ship's Ventilator, Looked Upon as a Solid. 

demonstration a separate diagram of triangles has been 
constructed, as shown at C D E. Here it will be noted 
that distances from B along the line B A have been set 
off from D along the line D C, thereby locating points 
which are at the same distance from a common base line, 
as D E. The lengths of broken lines as found in plan are 
set off from D along the line D E; when points are con- 
nected as shown to secure the true lengths of those lines, 



The Pattern, 

To develop the pattern we draw upon the plane of 
development a line whose length is found in the oblique 
line 1 1 of the elevation, and designate that end of the 



SHIP'S VENTILATOR 131 

line which shall be at the base of the pattern. The broken 
line 1 2 radiates from point 1 at the base, and is distant 
from point 1 at the top equal to the distance from 1 to 2 
of the true form of the top, therefore we set our com- 
passes to that distance, and with point 1 at the top of the 
pattern as center, describe a small arc as at 2. With 
compasses set to a span equal to the length of line 1 2 
found in the diagram of triangles, place one point at 1 
of the base, and describe a small arc as at 2 of the top. 

The intersection of these arcs, i.e., at 2 of the top, 
locates point 2 of the pattern in its correct relative posi- 
tion, and upon drawing the line 1 2 of the pattern, the 
triangle shown at 1 1 2 of the plan, has been constructed 
in its true form. We now add the triangle shown in 
plan at 1 2 2 by using the distance from 1 to 2 of the 
circle in plan as radius, and with point 1 at the base of 
the pattern as center to describe the small arc shown at 
2 , then the lower extremity of line 2 2 must lie in some 
point of this arc. 

To locate the exact position of that point, we set our 
compasses to the length of line 2 2 found in the diagram 
of triangles, and with point 2 at the top of the pattern as 
center, describe the second small arc as shown at 2 of the 
base. Lines may now be drawn to complete what may be 
termed one section of the pattern. The above operations 
continued, placing each length shown in the diagraih of 
triangles in its proper position, will complete the pattern 
as shown at Fig. 65. It may be well to again direct the 
reader's attention to the fact that the distances between 
the upper extremities of full lines are found in the semi- 
circle representing the true form of the top and not in the 
semi-ellipse shown in plan. 



132 TRIANGULATION 

On the Division of Circles. 

It is a difficult matter to lay down fixed rules for divid- 
ine circles to secure the best results: however, for the 
purpose of avoiding confusion, circles have been divided 
into sixteen parts throughout this work. In many in- 
stances this number is sufficient, especially if care be used 
when cutting the pattern, i.e., if the outline within which 
points are situated should be curved we may employ the 
eve to complete that curve. The old adage, the more 
points the more accuracy hardly holds good in a greater 
part of our work. In fact the writer has noted instances 
where it was folly to employ so many spaces, since some 
error committed was multiplied to that extent which ex- 
ceeded the slight discrepancies which would exist had a 
less number been employed. 

It is hardly worth while to consume additional time 
unless there is hope of increased accuracy. On the other 
hand, there are instances, as in this demonstration, where 
an increased number will no doubt increase the accuracy, 
since Ave presume the broken lines as 1 2,2 3, 3 4, etc., to 
be right lines, whereas if lines were drawn between these 
points upon the surface of the object they must become 
more or less curved, therefore error exists. More points 
of division would correct this to a considerable extent. 



CHAPTER XVII. 

On the Tapering Elbow to p.e Made in Any Number 

of Pieces. 

The tapering elbow is a fitting occasionally demanded. 
No doubt in some instances it is best developed by pre- 
suming it to be the frustum of a right cone which has 
been cut obliquely to its axis. This makes every piece of 
an equal flare throughout, therefore somewhat difficult 
to connect. 

Where the difference in the diameter of the end is not 
great, the tapering elbow is best developed as so many 
pieces of an elbow in round pipe, by gradually reducing 
the diameter of each piece, and compensating for this in 
the miter seams. There are occasionally instances where 
a departure from the above is desirable, when we may 
perhaps secure our patterns by triangulation. 

An attempt to show the relation between the ship's 
ventilator with a round mouth, and the tapering elbow, 
is somewhat a departure from fixed customs of the past, 
nevertheless there is close relation if we are allowed to 
modify the elbow slightly. It has been generally con- 
ceded that the right section of each piece in a tapering- 
elbow should be round. However, a slight variation 
from this would hardly be apparent in some of the larger 
work if it is made in a considerable number of pieces, i.e., 
five or more. 

The modification spoken of is to make each piece round 
at each end, and with some method of drawing a side ele- 
vation, a problem in close relation to the ship's ventilator 
is before us. The purpose of the writer is not to pro- 

133 



134 TRI ANGULATION 

long this work beyond a reasonable discussion of prin- 
ciples and methods which may be employed to secure the 
patterns for all forms where triangulation is to be ap- 
plied. Therefore he will simply attempt to show the rela- 
tion the tapering elbow may be made to bear to the ship's 
ventilator with a round mouth. 

Since the development of one piece of the ship's venti- 
lator has been explained, the reader should have little 
difficulty in securing patterns for the tapering elbow, be- 
yond drawing* the first diagrams to represent a side eleva- 
tion. In the specifications for a tapering elbow, we may 
find a fixed radius of throat, or a fixed radius of back, or 
it may be required to have an equal flare at back and 
throat, thus making a fixed radius for the center. 

Some Suggestions. 

The writer suggests methods as illustrated at Fig. 67 
for drawing elevations of tapering elbows. Here it has 
been presumed that the elbow is to be made in six pieces, 
and at an angle of 90 degrees. However, it will be 
readily understood that the number of pieces, or the re- 
quired angle, will make no material difference in the 
methods to be pursued, although the inclination of the 
miter lines will, of course, be dependent upon the angle 
and number of pieces. 

In No. 1 a given radius of throat has been assumed at 
A B, and the arc B C drawn with that radius. The miter 
lines are drawn of indefinite lengths at the same angle 
that would prevail for an elbow of a constant diameter. 
Using the same methods that would be employed for an 
elbow of a constant diameter, the elevations of the pieces 
at the extremities of the tapering elbow are drawn in 
positions as shown, thus establishing the lengths of two 
miter lines, i.e., m o and n p. 



ON THE TAPERING ELBOW 135 

To secure a symmetrical form it is fair to presume that 
the remaining miter lines, i.e., d e, f g, and h k, should be 
of proportionate lengths. These proportionate lengths 
may be secured in many ways, one of which is shown at 
No. 4, Fig. 67. In No. 4 the line D E is drawn to a 
length equal to the length of miter line n p, and a distance 
set off from E equal to the length of miter line m o, as at 
F. The line F D is divided into as many parts as there 
are remaining pieces in the elbow, thus securing points 
as 1, 2, and 3. The upper extremities of the miter lines 
in No. 1 are now located by setting off from k along the 
miter line k h, a distance equal to E 1 in No. 4; from g 
along the miter line g f a distance equal to E 2, and from 
e along the line e d } a distance equal to E 3. With points 
connected as shown at No. 1, an elevation is completed, 
when a given radius of throat is demanded. 

In No. 2, Fig. 67, a given radius of back is assumed as 
G H, and an arc drawn as shown. The elevations for the 
pieces at the extremities of the elbow, and the miter lines, 
are drawn in the same manner as was explained for No. 
1. Here it will be apparent that the lower extremities 
of the miter lines as at s, t, and u, may be located in posi- 
tions which suit our fancy, or, in other words, in positions 
which will give the fitting the best form when finished. 

In No. 3, a given radius of center is assumed as X Y, 
and the arc Y W drawn as shown. The elevations for 
the pieces at the extremities of the elbow, and the miter 
lines are again drawn in positions as shown, and in the 
same manner as has been suggested for Nos. 1 and 2. 
Here we use the same lengths of miter lines as was used 
in No. 1, although points on the arc Y W are looked upon 
as the centers of those lines, i.e., we set off one-half the 
length of each on either side of the arc Y IV. By draw- 
ing lines to connect points which have been located in 



136 



TRIANGULATIOX 



this manner upon the miter lines as shown, an elevation 
is completed. 

It will be noted that this gives the fitting a symmetrical 




Fig. 67. Side Elevations of Tapering Elbozvs. 

form when viewed from the side, and allows the ends to 
be comparatively straight for connecting. There is, of 
course, some slight distortion in these pieces, which is 



ON THE TAPERING ELBOW 137 

the outcome of making both ends round when not 
parallel. 

Assuming that each piece in the elbows whose eleva- 
tions are shown in Fig. 67 is to be round at each end, and 
of diameters equal to the lengths of lines which represent 
those ends in elevation, we have an exact counterpart of 
the ship's ventilator when a formula is supplied for a side 
elevation of that object. To secure the pattern we may 
proceed in precisely the same manner as has been ex- 
plained for the ship's ventilator. 

In discussing this matter with a man familiar with this 
class of work, the question was raised : "What shall we 
do if it is required to make the elbow straight on one 
side?" The reply was: "If you wish to secure the pat- 
tern for an elbow of this class when the ends are what 
you call 'off center' construct your plans accordingly. 
This will demand a full plan for each piece, as one-half 
is not a duplicate of the other. Therefore the complete 
pattern for each piece must be developed." 

To Draw tlie Plan When the Elbow is to be 
Straight on One Side. 

Fig. 68 is shown in an endeavor to illustrate methods 
which may be employed to draw the plan when a fitting 
of this class is required, which is commonly known as 
"straight on one side." It is, in fact, presumed to repre- 
sent the piece marked A in the elevation for the ship's 
ventilator, if it was required to make that piece straight 
on the side furthest from the eye, and with the form and 
diameter of the ends remaining the same. As will be 
noted, the elevation is substantially the same as shown 
in Fig. 65, Chapter XVI. 

For the plan a circle is drawn equal in diameter to the 
length of the base line and in a position as indicated by 



138 



TRIANGULATION 



the vertical projectors. We may now presume the oblique 
line X Y of the elevation Fig. 68 to be the intersecting 
line between the vertical plane and a supplementary 
oblique plane. 

Draw in position as shown a circle whose diameter is 




Fig. 68. Methods That May Be Employed to Draw the Plans 
When a Tapering Elbozv Is Required to Be "Straight on One. 
Side." 

equal to the length of the oblique line X Y, using care to 
place it at the same distance above the line X Y as the 
large circle in plan has been placed below the line / L. 
This represents the true form of the top. Divide each 



ON THE TAPERING ELBOW 139 

circle into the same number of equal parts as shown. 
From the points of division of the circle which represents 
the true form of the top, project lines to intersect the 
oblique line X Y as shown at points a, b, c, d, etc. From 
these points of intersection, i.e., a, b, c, d, etc., draw in- 
definite lines below and perpendicular to the line / L. 
Points as a, b, c, d, etc., may be looked upon as the end 
elevations of lines which cross the top. 

The plans of these lines are some portions of the per- 
pendicular lines drawn below the line / L from said 
points. Since the lengths of these lines are shown in the 
lines which cross the circle representing the true form 
of the top, it only remains to locate their extremities in 
their correct relative positions. 

It will be noted that point d is in reality an elevation 
of the line designated as 5 13 upon the true form of the 
top. The extremities of this line are distant from the 
vertical plane equal to the lengths of lines d 5 and d 13 
of the true form of the top. Therefore if we set off below 
the line / L upon line shown at 5 13 of the plan, distances 
as found from d to 5 and d to 13 of the true form of the 
top, those points are located in plan as shown. Continue 
this operation until each point has been located in plan. 

A line traced through these points forms an ellipse, 
which is a plan of the top and may be numbered as shown. 
Draw lines between points of the same number located 
upon the circle and ellipse in plan, and plans of full lines 
presumed to be upon the surface of the object are secured. 
A\ lien broken lines are drawn as 1 2, 2 3, 16 IS, 15 14, 
etc., plans of those lines are also secured. 

It may be explained that to avoid confusion in the 
elevation, lines have been drawn in such a manner as to 
allow one line in elevation to represent two in plan. Hav- 
ing now before us the plan and elevation for each line 



140 TRI ANGULATION 

presumed to be upon the surface of the object, we con- 
struct triangles to secure their true lengths in the same 
manner as was explained for the ship's ventilator. Lines 
placed upon the plane of development in lengths as 
found in the diagram of triangles so constructed, and in 
their correct relative positions, supply points through 
which lines are traced to secure the pattern. 



CHAPTER XVIII. 
Transitional Elbow in Rectangular Pipe. 

There is a demand for the transitional elbow in rect- 
angular pipe in some branches of sheet metal work. To 
satisfy this demand, passable results may be secured by 
applying triangulation to the development of its patterns, 
although the author has never seen an example wherein 
ideal results were obtained when the throat and heel 
were cylindrical. 

This can be attributed to the fact that a portion of 
such forms is in close relation to the form known as the 
Right Helicoid. The Right Helicoid is a warped sur- 
face, and cannot be obtained without a drawing or 
stretching of the material when made from sheet metal. 

In the following demonstration it has been presumed 
that one side of the elbow, or that which is commonly 
known as one cheek, is to be flat. This has been the 
case in nine out of ten examples which have come to the 
author's notice. 

Plan and Elevation of the Elbow. 

Fig. 69 shows the plan and elevation of an elbow of 
this class. Here it will be noted that the throat and heel 
have been cut obliquely, as shown by lines A B and A C 
in elevation. The elevation of a short collar at one end 
is shown by DAE F, and B C G H is the collar when 
looking into the other end. The plan clearly shows the 
throat and heel. 

To secure the true lengths of lines presumed to be 
upon those parts, we divide the curved portion of said 

141 



142 



TRIANGULATION 








^ 


i 






/ § 


c 


1 1 ^ 






// *= 


«y 


// & 




/,/ r 




~» 


atter 




f *" 


^c 








fei 



CtJ 



TRANSITIONAL ELBOW 143 

lines into convenient parts as shown at T 4 5 6 0, and 
K 1 2 3 M. From these points of division lines are pro- 
jected to intersect the oblique lines A B and A C of the 
elevation, as also shown. The patterns for the throat and 
heel are now secured as for any cylindrical form which 
has been cut obliquely, i.e., in the same manner that the 
patterns are secured for an elbow in round pipe. In other 
words, the above spoken of points of division may be 
looked upon as plans of elements of the cylindrical sur- 
face, and since the plan supplies the distance between 
these elements, we have only to determine their lengths 
to develop the patterns for the throat and heel as shown. 

The points of division previously located upon the arcs 
in plan are also utilized as points between which lines are 
drawn, and presumed to divide the surface of the upper 
cheek, or top, into triangles. The elevation clearly shows 
these lines, although in a problem of this class, they are 
by no means necessary, since measurements may be se- 
cured from the patterns for the throat and heel. These 
lines in elevation may at times be an element in avoiding 
confusion, and may also be utilized as here shown, to 
determine the difference in bight of the extremities of 
lines upon the surface of the elbow of which they are the 
elevation. 

True Lengths of Lines Upon the Top. 

Presuming the patterns for the throat and heel to 
have been secured as shown at Fig. 69, our next work is 
to determine the true lengths of those lines which cross 
the upper cheek, and shown in plan at T K,T 1,T 2,T 3, 
T 71/, also M 4, M 5, M 6, and M 0. This is accomplished 
by the use of the right angled triangle as shown in the 
diagram of triangles Fig. 69, since from the plan we se- 
cure the length of base for each triangle, and from the 
elevation the perpendicular is secured. 



144 



TRTANGULATION 



It may be here explained that in case the work is large, 
and it is desirable to avoid making a plan and elevation, 
we may look upon the lower cheek as a plan, and draw 
lines upon that surface which shall represent the tri- 
angles presumed to be upon the opposite cheek. From 
the patterns for the throat and heel we can determine the 
difference in hight of the extremities of those lines, 
thereby securing the perpendiculars for all triangles. 




Fig. 70. Sccno graphic Representation of Elbow in Rectangular 

Pipe. 

These triangles may of course be drawn upon the sur- 
face which constitutes the lower cheek, although in this 
we have constructed separate diagrams. To secure a 
better understanding of this, the reader may construct 
his elbow as the patterns are secured, thereby placing a 
model before him. 



Pattern for Upper Cheek or Top. 

The pattern for the upper cheek is secured by placing 
lines in their true lengths and positions upon the plane 
of development, as shown in the pattern for the top, Fig. 
69. As for example, the surface shown in plan at b K T 
is triangular; b K is the true length of one side of the 
triangle, and the true length of b T is secured either from 
the elevation, or from the pattern for the heel. The 
true length of line T K is secured from the diagram of 



TRANSITIONAL ELBOW 



145 



triangles. The true lengths of the four remaining lines 
radiating from point T are also found in the diagram 
of triangles. The distance these lines are from each 
other at the throat is secured from the pattern for that 
portion. By applying the same reasoning to those lines 




Fig. 71. Parts of Object to Be Constructed as An Experiment. 

radiating from point M, it is but a simple operation to 
develop the pattern for the upper cheek as shown. 

Variation in Methods. 
The work of securing the patterns for an elbow of ibis 
class may be somewhat simplified by cutting the upper 
edge of fehe throat and heel upon straight lines, as shown 



146 TRIANGULATION 

by broken lines a x and d y of the patterns for the throat 
and heel, thereby dispensing with an elevation. When 
this course is pursued, the method of developing the pat- 
tern for the upper cheek differs in no material respect, 
since lines may be located upon those portions which are 
to form the throat and heel, whose plans will be points 
Kl 2 3 M <mdT4 5 60 of the plan as shown. 

Breaks or Bends in the Upper Cheek or Top. 

Either mode of procedure demands that there be breaks 
or bends in the material upon lines A D, A B, and B C 
shown in the scenographic representation of an elbow in 



Fig, 72. Object Constructed from Parts Shown in Fig. 71. 

Fisr. 70. These breaks or bends are the objectionable 
feature, although difficult to eliminate, especially if the 
rise is considerable, from the fact that the surface re- 
sembles the above spoken of surface, the Right Helicoid. 

A Simple Experiment. 

If the reader is of an experimental turn of mind, and 
wishes to prove beyond question the truth of the above 
statement, he may draw two concentric arcs as shown at 
A Fig. 71, and cut from sheet metal two triangular pieces 



TRANSITIONAL ELBOW 147 

whose base lengths are equal to the length of the arcs, 
and whose perpendiculars are equal as shown at E and F, 
Fig. 71. Form these triangular pieces so that their bases 
will conform to the arcs shown at A, and construct an 
object as illustrated at Fig. 72. He may then use any 
flexible but non-elastic material to cover the space be- 
tween the two cylindrical forms, and endeavor to fit it 
to the upper edge of each at the same time, thereby 
supplying a surface known as the Right Flelicoid. 

The author has found it a difficult matter to convince 
the average man that this surface is warped and cannot 
be developed without a stretching or drawing of the 
material. He therefore suggests the above experiment 
as a proof that this surface cannot be developed abso- 
lutely, even though triangulation be applied. 



CHAPTER XIX. 

A Transitional Elbow from Round to Elliptical. 

Herein will be discussed methods to secure the patterns 
for an elbow as shown in a pictorial way in Fig. 73, i.e., a 
four pieced elbow from round to elliptical. 

This problem is closely related to others previously ex- 




Fig. 73. Transitional Elbow from Round to Elliptical. 

plained, although there are a few new features involved. 
The problem is not a difficult one beyond the fact that it 

148 



ROUND TO ELLIPTICAL 149 

is somewhat prolonged, since there are four separate and 
distinct patterns to produce. 

J t may be well to here explain that no doubt in practical 
work, the demand will be for an elbow made in five, six 
or seven pieces. 

This work has been designed to illustrate and explain 
the principles involved. Therefore the illustrations have 
been reduced to as simple examples as are consistent with 
the problems in hand. The author's sole object being to 
enlighten rather than confuse. 

Since the principles involved would be the same re- 
gardless of the number of pieces, he trusts that those 
principles may be best comprehended from the more 
simple examples. Moreover, if one succeeds in securing 
the patterns for an elbow of this class made in four pieces, 
he will have little difficulty in securing the patterns for 
the elbow made in a greater number of pieces. 

In the following example, it has been presumed that 
the specification demands an elbow of 90 degrees to make 
connection between a 16-inch round pipe, and an elliptical 
pipe, whose major and minor diameters are 24 and 12 
inches respectively. A scale has been appended in Fig. 
74 to enable the reader to more readily follow the work 
by comparing measurements. 

In a problem of this class, a complete plan of the elbow 
entails a considerable outlay of labor, and is unnecessary. 
Therefore it has been omitted. Beyond this, it has been 
presumed in this example that the axis of each piece is 
in one plane, or what is commonly known as "on center". 
Thus it is only necessary to draw one-half of each profile, 
as all semi-patterns may be duplicated for the other half. 

As will be noted in Fig. 74, a semi-circle has been 
drawn as shown at A, the diameter of which is equal to 
the diameter of the round pipe, or 16 inches. At any 



150 



TRT ANGULATION 



convenient distance above the line / L draw a line as 
shown, or at a distance equal to the required length of the 
round collar as B C. Presuming the required radius of 




i|iii|i|i|i|imiii|i|iiiiiimii|imi 



TIT 



I'l'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'ITI'I'I'I'ITI'l 1 ! 1 ! 1 ! 1 !' 

9 12 15 18 21 2 
Scale . 



Fig. 74. Side Elevation and True Form of Ends of Elbow in 
Fig. 73. 



throat is 9 inches, then point B is the center from which 
an arc is drawn, the radius of which is 9 inches, as shown 
at E D. We may now treat the arc E D in the same 
manner that a similar arc would be treated for an elbow 



ROUND TO ELLIPTICAL 151 

of a constant diameter, as shown at a, b and c, i.e., locate 
points through which miter lines would be drawn pre- 
suming the elbow was to be a four-pieced elbow in 16-inch 
round pipe. 

Through these points of division, draw lines as shown 
at B a F, B b G, and B c FI, which will represent the 
miter lines of the elbow. From point F draw the line as 
shown at E J. Above this line draw a semi-profile of the 
elliptical end of the elbow with the major axis at right 
angles to line F J as shown at T. 

The arc 1 9 is divided into an equal number of equal 
parts as shown at 1, 2, 3, 4, etc., and lines projected from 
said points of division to intersect the miter line c H, 
thus securing an elevation of the round collar, or the 
true lengths of the rectilinear elements of the cylindrical 
surface. From this the pattern may be secured as for 
any elbow the diameter of which is constant. In like 
manner the semi-profile of the elliptical end is divided 
into an equal number of equal parts as shown at T, and 
lines projected to intersect the miter line a F as shown. 
This supplies two views of the elliptical collar, and the 
pattern for that portion may be secured as recommended 
for all parallel forms. 

To Establish a Form for an Intermediate Section. 

When following this mode of constructing an elbow, 
we have two cylinders, one circular and one elliptic, 
which have been cut obliquely, and forming two of its sec- 
tions. Since the two intermediate sections are to con- 
nect these upon the miter lines c II and a F Fig. 74, it 
follows that one end of each must be of a suitable form 
and size to make said connections. Therefore in this 
example, the only unknown form is that upon the miter 
line b G. 



152 TRT ANGULATION 

This form can be arbitrarily established. However, it 
is not to be recommended, since it is a somewhat difficult 
undertaking to establish a form which will be productive 
of satisfactory results in the finished elbow. The author 
suggests that a form be found which will be in propor- 
tion to the two forms previously established upon the 
miter lines a F and c H. 

The two diameters of this, the required ellipse for a 
suitable form of the elbow on line b G, may be secured by 
graphical methods as was explained for a similar ex- 
ample in the seventeenth chapter, or as it may be arrived 
at somewhat as follows : — We find that the ellipse which 
is the true form of the right circular cylinder upon line 
c H, will have dimensions of approximately 16j/2 and 16 
inches as the major and minor diameters. In like manner 
we find that the ellipse which is the true form of the right 
elliptic cylinder upon miter line a F , has dimensions of 
approximately 12 and 24*4 inches. The difference then 
in the two major diameters is 8)4 inches. We may divide 
this difference in this case by 2 which gives us 4^s inches. 
This maybe either added to \6y 2 or deducted from 24^4, 
which gives us 20^ inches as the major diameter of the 
required ellipse, which we shall presume to be the true 
form upon line b G. 

In like manner we find that the difference between 12 
and 16 is 4 inches. This divided by 2 is 2, which added 
to 12 or deducted from 16 equals 14 inches, the minor 
diameter of that ellipse. We can now definitely locate 
the point h as shown upon the miter line B b h G, since 
it will be 20^8 inches from point b as shown at h. Since 
the forms for each end of the two intermediate sections 
are elliptical, and it is necessary that these forms be 
drawn to secure the patterns for these sections, we could 
emplov any convenient method for securing them, 



ROUND TO ELLIPTICAL 



153 



The Ellipse. 

There are many ways of describing that curve known 
as the ellipse, and it makes no material difference how it 
is secured. It may be well to here explain that in the 
strictest sense, no part of an ellipse is a part of a circle. 
Therefore methods recommended for what is known as 
the false ellipse, i.e., those drawn from centers, will 
be somewhat in error, although the variation in many 




Fig. 75. Methods of Securing 
True Form of Oblique Section 
of Cylinder. 

instances will be insignificant and may be consistently 
ignored. On the other hand, the author believes that the 
more accurate methods of securing that curve are to be 
desired. 

As for example, the oblique section of a right circular 
cylinder is an ellipse, the diameters of which are depen- 
dent upon the diameter of the cylinder, and the angle at 
which said cylinder is presumed to be cut. By keeping 
this in mind, we are always prepared to locate points in 
an ellipse of given dimensions, as will be explained. 



154 TRIANGULATIOX 

The true form upon miter line b h, Fig. 74, in this ex- 
ample, is an ellipse with diameters of 14 and 20^ inches. 
To secure this form we may draw a semi-circle, the di- 
ameter of which is 14 inches, as shown at Fig. 75, and 
divide said semi-circle into a convenient number of equal 
parts as shown at 1, 2, 3, 4, etc. From these points of 
division project lines at right angles to line 1 9, as also 
shown. 

Thus we have before us the plan and elevation of a 
semi-cylinder which we shall presume to cut at a suitable 
angle to give us a length of 20^ inches for the section 
line, as A B. Points as a, b, c, d, etc., thus secured, are 
looked upon as the end elevations of lines which cross 
the semi-cylinder, and the horizontal projectors from 
points 2, 3, 4, etc., i.e., from said points to the line 1 9, 
are the plans of those lines, and are in this instance, their 
true lengths. 

Draw lines from points a, b, c, etc., perpendicular to 
line A B. Set off distances upon said lines as found in 
plan, as for example, the distance from m to 3 is set off 
from b on line b X, and so on for all lines shown. We. 
have thus secured points in the required ellipse. To 
facilitate our work, we may cut a templet from this, for 
the purpose of duplicating this curve whenever it becomes 
necessary in diagrams which must be drawn to secure the 
patterns for the two intermediate sections. 

Since the semi-patterns for either intermediate section 
are secured by duplicate operations, but using somewhat 
different measurements, one only is discussed in this 
demonstration, i.e., that included between points c, b, h, 
H, Fig. 74. The form of one end of this section is ellip- 
tical, as shown at Fig. 75. Therefore in constructing the 
necessary diagrams to secure its pattern, we may first 
draw the semi-ellipse 1,2,3,4, etc., as shown at Fig. 76, 
and look upon this as a plan. 



ROUND TO ELLIPTICAL 155 

Above this as shown, we duplicate the elevation of 
section c b li 11, Fig". 74, by first presuming the line 1 9 
to be an elevation of what we may now term the base. To 
locate the remaining points, i.e., c and II, in their correct 
relative positions, we may divide our primitive elevation, 
Fig. 74, into triangles as shown by the broken line c h, 
and construct similar triangles at Fig. 76, thereby secur- 
ing a duplicate elevation in a somewhat changed position. 

The necessary form of the elbow upon line c H, Fig. 74, 
is also elliptical, and may be secured by determining the 
true form of the 16-inch round pipe upon line c H. This 
form drawn in a position as shown at a e i, Fig. 76, 
supplies a plan of the base, an elevation, and the true form 
of the top. The true form of the top as here represented, 
is presumed to be upon an oblique supplementary plane, 
and the line a i is the intersecting line between this, and 
the vertical plane. 

Plan, Elevation and Diagram of Triangles. 

An examination of the construction lines shown in Fig. 
76 should render the work of drawing a complete plan 
and elevation of the semi-section a simple matter. 

As will be noted, horizontal lines are drawn from 
points on line a i, which are at a distance from / L equal 
to the vertical bight for all triangles. From the base of 
some vertical line as at Y, we may set off distances as 
found in the lengths of full lines drawn in the usual man- 
ner between similar points in plan. From the base of 
some vertical line as at X, we also set off distances as 
found in the lengths of the indirect, or broken lines which 
have also been drawn in the usual manner. Lines are 
drawn from points thus located, to the respective inter- 
sections of horizontal lines and the lines extending from 



156 



TRIANGULATION 



-J 



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Ss 



Ss 



/, 








h-H 



v a 



Q 



a Co 



Bh 



■ *» 
ft. 



ROUND TO ELLIPTICAL 157 

Y and X, thereby determining the true lengths of lines 
shown in plan. 

Thus in the diagram of triangles we have the full 
lengths of necessary lines shown in plan, and from the 
plan and the true form of the top, we determine the 
distances said lines are from each other at their ex- 
tremities. 

The Pattern. 

To secure the pattern as shown, we draw a line in any 
convenient position upon the plane of development, the 
length of which is a 1 of the elevation, as a 1 of the pat- 
tern, Fig. 76. Line 2 b is distant from a 1 at the lower 
extremity equal to the distance between 1 and 2 of the 
plan, and at the upper extremity equal to the distance be- 
tween a and b of the true form of the top. 

To locate these points upon the plane of development, 
we use points a and 1 of the pattern to describe arcs, the 
radii of which are equal to the distances said lines are 
from each other; then the extremities of line 2 b must 
lie in some points of these arcs. We find from the dia- 
gram of triangles that the true distance from 1 at the 
base to b at the top, is the length of the broken line 1 b 
in the diagram of triangles. 

Using this as radius and with point 1 of the pattern as 
center, we describe the second small arc as at b, thereby 
definitely locating the upper extremity of line 2 b upon 
the plane of development. Since the true length of line 
2 b is also found in the diagram of triangles, we use that 
as radius, and with point b as center, describe the second 
small arc as at 2, thereby completing what may be looked 
upon as one section of the semi-pattern. With these 
operations repeated for each section shown in plan, and 
using proper distances as found in the several diagrams, 
the pattern is completed as shown. 



158 TRI ANGULATION 

Should the student become interested in securing pat- 
terns for a form of this nature which is required to be 
"off center" or flat on one side, it may prove to his ad- 
vantage to re-read Chapter XVII. 

Application of the So-called Rule of Thumb. 

It may at least be interesting to note that the so-called 
rule of thumb could have been introduced to secure the 
pattern by mechanical methods, by cutting from sheet 
metal a form as shown at Fig. 76, i.e., that bounded by 
line 1 a e i 9 and 5, then bend upon lines a i and 1 9 to 
form an object as shown at Fig. 77 . If a piece of paper 
be wrapped about the curved face of this and marked, the 




Fig. 77. Object from Which Pattern May Be Secured. 

pattern sought is secured, providing of course, that the 
semi-elliptical fprms are at right angles to what may now 
be termed the back. 

This is explained in the fact that all sheet metal pat- 
terns are what may be looked upon as the envelope of a 
solid, the form of which is that of tlie required object. 



ROUND TO ELLIPTICAL 159 

Triangulation is but the process of measuring- these sur- 
faces, and applying' these measurements to the plane of 
development for the purpose of locating points in the 

outline which hounds the pattern. 



CHAPTER XX. 

The Helical Elbow. 

The helical elbow is somewhat of a novelty in the sheet 
metal industry, and can hardly be recommended for 
general work. Other forms of ducts may usually be 
designed which will fulfil the requirements to better ad- 
vantage. On the other hand, we may meet with an 
insistent demand for a fitting of this class. Therefore it 
seems desirable to discuss it in this work, although per- 
haps it makes the rule somewhat elastic when placed 
under the head of Triangulation. 

Fig. 78 shows in a pictorial way a helical elbow ex- 
posed in the corner of a room and presumed to make con- 
nection between a pipe or duct passing through wall A 
and a similar pipe passing through wall B, which is at 
right angles to wall A. The pipe or duct in wall B is at 
a greater distance from the floor than that in wall A, 
therefore a considerable rise or pitch is demanded in the 
elbow while making a revolution of 90 degrees. In other 
words, the pipe or duct is required to revolve about the 
corner of the room as an axis and to have an equal rise 
for every unit of revolution. The heel and throat are 
but portions of a right circular cylinder, and the top and 
bottom or two cheeks, should be that surface known as 
the right helicoid. 

It has been previously stated that the right helicoid 
is a warped surface and cannot be developed or forced 
into shape without a drawing or stretching of the 
material. On the other hand, we can secure passable 

160 



THE IIKL1CAI. I-ILI'.OW 



161 



results by introducing a series of breaks or bends, as will 
be hereinafter shown. Measurements are shown in Fig. 
7S, and the following- diagrams have been worked to 
those measurements by using the scale in Fig. 79. It may 
be well to remind the reader that the distances in Fig 78 




Fig. 78. Scowgraphic Representation of Helical Elbow in 
a Room Corner. 

cannot be compared with the scale, since that drawing is 
purely scenographic. 

From measurements shown in Fig. 78 it is but a simple 
operation to draw a plan. As for example, we draw a 
right line as C D, Fig. 79, and look upon this as a plan 
of wall A } Fig. 78. A line drawn from D perpendicular 



162 TRIANGULATION 

to C D, as D E, may likewise be looked upon as a plan 
of wall B. Then the point D is the plan of the vertex of 
an angle formed by the two walls of the room, or the axis 
of the elbow. 

On examination of Fig. 78 we note that each end of 
the elbow is 15 inches distant from the corner of the 
room. Therefore we use that distance as radius and 
with point D, Fig. 79, as center, describe an arc of 90 
degrees, as shown at F G. This arc is then looked upon 
as a plan of the throat. With the same point as center 
and with trummels set to a span of 45 inches as shown 
in Fig. 78, we describe the larger arc as also shown in 
Fig. 79. We have then before us the plan of the required 
elbow. 

From the specification of the elbow, i.e., it must have a 
gradual rise throughout its 90 degrees of revolution, we 
conclude that the upper and lower edges of the throat and 
heel must describe in space that form known as the 
helix.* From the definition of the helix as given in the 
note below, we may also conclude that any right line 
drawn obliquely across the envelope of a right cylinder, 
will, when said envelope is developed into a cylinder, de- 
scribe a helix in space. The angle at which this line 
should be drawn is dependent entirely upon the required 
rise or pitch of the helix, in the whole or part of a revolu- 
tion. 

Patterns for Throat and Heel. 

Presuming the reader has acquired an understanding 
of the above, we may now proceed to secure patterns for 
the throat and heel by representing upon the plane of 

*The Helix is designated as the path of a point, which, while revolving 
uniformly around an axis, also moves uniformly in a direction parallel 
thereto. This curve then lies upon the surface of a cylinder, cuts all its 
rectilinear elements at the same angle, and becomes a right line when the 
cylinder is developed into a plane. 



THE 



ELICAL ELBOW 



163 





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V X \ *"*X 












X \ ° X 






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x 5" 11 \ 














\ X f* X 








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v \ S 


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164 TRIANGULATION 

development the envelopment of the cylindrical forms, the 
plans of which are shown in Fig. 79. This is accomplished 
by drawing right lines equal in length to the lengths of 
those arcs, and through the extremities of said lines 
drawing additional lines perpendicular for the first, as 
shown at H I and J, Fig. 79. To confine Fig. 79 within 
reasonable bounds, one pattern has been made to overlap 
the other to some extent. This necessitates one being 
transferred when the pattern is cut. 

As for example, the line H J is equal in length to arc 
C E, and a portion of the same line as / /, is the length 
of arc F G. We look upon the surface H K L J as the 
envelopment of a cylindrical form the plan of which is 
arc F G. We note from Fig. 78 that the rise of the elbow 
is to be 24 inches in its revolution of 90 degrees. There- 
fore we locate a point 24 inches from / upon line / L as at 
0. Lines drawn from points H and I to O, then repre- 
sent the lower edges of the throat and heel upon the pat- 
tern. Since the pipe to be connected is shown to be 15 
inches high, we locate points as P R and S, which are 
15 inches above points H I and O. On drawing lines as 
P S and R S, the patterns for the throat and heel are 
completed, as shown. 

We may here explain that when following this course 
to secure a connection as shown in Fig. 78, some distor- 
tion will exist together with some reduction in the 
capacity of the duct. This distortion may be modified, 
and the capacity increased, as will be hereinafter ex- 
plained. For the present we shall presume that the elbow 
is to be made precisely as shown in Figs. 78 and 79. 

Patterns for the Top and Bottom. 

To secure the patterns for the top and bottom of the 
elbow, we may divide the arc C E, Fig. 79, into four equal 



THE HELICAL ELBOW 165 

parts, and sub-divide the first division from C into 3 or 
more equal parts as b, c, and d. By the use of our 
straight edge we draw lines as b c, c f, and d g, which 
are divergent lines from point D, and cut the arc F G 
in points c f and g. Points as b c and d upon arc C E, 
are now looked upon as the plans of elements upon the 
above spoken of cylindrical surface, of which K H L J 
is the covering. Therefore we may locate one or more 
points upon line H J , with distances between as found 
between c b, b c, or c d of the plan, as shown at 1, 2 and 3. 
Lines are drawn perpendicular to said lines as shown, 
and intersecting those lines which represent the top and 
bottom of the pattern for the heel of the elbow. 

In like manner we locate points along the line / / with 
distances between as found between points F and e, etc., 
on arc F G of the plan, as shown at 4, 5, and 6. From said 
points perpendicular lines are erected to intersect lines 
R S and / O, which are the top and bottom boundary 
lines of the pattern for the throat. . The true distance be- 
tween points, the plans of which are C, b, c, etc., is then 
found between points P and y of the pattern. Likewise 
we find the true distance between points, the plans of 
which are at F, e, f, etc., between points R and w of the 
pattern for the throat. 

All divergent lines from point D shown in the plan 
are perpendicular to the corner of the room, or horizontal. 
Therefore broken lines as C c, b f, etc., of the plan, are in 
reality at an angle to the horizontal, dependent upon the 
rise of the elbow between those points, i.e., that vertical 
distance as shown between P and X of the pattern. To 
secure true lengths of broken lines shown in plan, we con- 
struct a triangle as shown at No. 1, Fig. 79, with base 
equal in length to line C c of the plan, and the perpen- 
dicular of which is P X of the pattern. Then U V is 



166 TRI ANGULATION 

the true length of not only C c of the plan, but of all 
similar lines shown or assumed. 

Having before us the lengths of all lines which we pre- 
sume to be upon the pattern, and as the pattern is but a 
series of triangles joined together, the dimensions of 
which are equal, we can, if thought more convenient, cut 
one section from sheet metal, the plan of which is C F c b. 
Fig. 79, and shown at C F e b of the pattern, Fig. 80. 
This section may be duplicated upon the plane of develop- 
ment to complete the whole pattern, or we may use our 
trammels in the usual manner. However, since in this 
instance the pattern is composed of twelve equal sections, 
all similar measurements will be equal, as is shown at the 
pattern, Fig. 80. Care must be used in this to secure 
accurate lengths for the throat and heel, which should be 
equal to those lengths shown at P S and R S, Fig. 79. 
As is indicated at Fig. 80, the inside of the top cheek is 
shown. 

The full and broken lines which bound the several 
triangles of which this piece is composed, are also a key 
to the direction the metal should be bent. That is, the 
metal is to be bent up on the broken lines, and down on 
the full lines. The angle of these bends is dependent 
upon the rise of the elbow and the radius of the throat, 
i.e., the radius of its plan. As for example, we look upon 
the surface C F e b, Fig. 79, as a plane which may be re- 
volved about the line C F as an axis. Presuming this to 
have been revolved in a manner to elevate point e 1 inch 
above point F, then point b would be considerably more 
than 1 inch above point C. However, since in reality the 
elbow is no higher at point b than it is at point e, a bend 
is necessary upon line C e. 

We could as consistently assume a bend on a line 
drawn from F to b, since it would make no material dif- 



THE HELICAL ELBOW 



167 



ference except to demand the bend in the opposite direc- 
tion. Thus it is but a simple matter to establish in theory 
the angle at which the metal should be bent upon those 
lines. However, since this bending process serves our 
purpose in another way, the exact angle in every instance 




Fig. 



Pattern for Top and Bottom of Elbow. 



has little or no effect upon the required results. There is 
a super-abundance of material in the center of the cheek, 
and by making these bends, and drawing out the edges 
so as to make the cheek appear somewhat as shown at 
l ; ig. 81, the required twist is developed. Too little twist 
denotes that there has not been a sufficient quantity of 
material consumed in the center, and too much twist de- 
notes that too much material has been consumed. There- 



168 



TRI ANGULATION 



fore we must make the bends shallower or deeper as the 
case demands. 

Loss of Capacity When Used as a Duct. 

Some loss of capacity will be found in an elbow of this 
class,, as well as some distortion, where said elbow makes 
connection with ducts of a given area of cross-section. 
As for example, in this instance it has been presumed 
that the elbow is to connect ducts with cross-sections of 
15 x 30 inches. We find upon referring to Fig. 79 that 




Fig. 81. Pictorial Representation of Top or Bottom, Show- 
ing Breaks or Bends. 

the width of the heel is approximately 14 inches and of 
the throat approximately 10j/2 inches. This, as will be 
noted, makes an average of 12j4 inches, or in a primitive 
way, w r e find that the elbow will average 12}4 x 30 inches. 
Fig. 82 is included for the purpose of conveying to the 
reader- through the medium of the eye, an understanding 
of this Upon examination, it will be noted that this 
diagram shows in a pictorial way, portions of the heel 
and throat of the elbow connected to the duct. The heel 
and throat are in reality strips of material cut at the ends 
at a suitable angle to secure the required rise in a given 



THE HELICAL KLUOVV 



169 



length. As for example, the heel and throat must rise 
to the same level while passing around cylindrical forms 
of varying diameter. Therefore that which passes 
around the cylindrical form of the least diameter must 
have the most rise per unit of measurement upon the base 
of the cylinder. Thus in every instance the throat must 
have a greater relative rise than the heel. Since vertical 
lines upon each are of equal lengths, the material which 
forms these portions must be of varying widths, de- 
pendent upon the radius of those parts. 

The Right Helicoid.* 

Fig. 84 has been constructed to conform to the defini- 
tion of a right helicoid, which was given in the eighteenth 




Fig. 82. Portions of Throat and Heel Connected to Duct. 

chapter, also in the note below. The vertical line E is 
presumed to be the vertical axis. The thirteen horizontal 
lines are presumed to be one line which has been revolved 



* The right helicoid is a surface which may be conceived as being 
generated by a right line revolving about an axis, and perpendicular to it ; 
also moving uniformly parallel to said axis. A surface so generated is 
usually presumed to lie between two concentric cylinders. 



170 



TRIANGULATION 



about the vertical line E, and shown in different positions 
which are 1]/2 degrees distant from each other while 
making - a quarter revolution. 

Presuming" the total rise to be 24 inches, the rise for 




Fig. 83. Duct Enlarged at Intersection of Duct and Elbow. 




Fig. 84. Generation of Surface Known as Right-Helicoid. 



each position would then be 2 inches. Arrow heads 
point out the plan of the path which the right hand ex- 
tremity of this line would describe in space while being 



THE HELICAL ELBOW 171 

revolved. From this we conclude that with a given rise 
for the elbow, as the throat and heel approach the axial 
line the pitch increases, and conversely, as these parts 
recede from the axial line the pitch decreases, i.e., the 
relative pitch to the base of cylinders of which they may 
be conceived as being a part. When the radius is Zero, 
the throat becomes a vertical line. As the throat of the 
elbow recedes from the axial line, loss of capacity and 
distortion decrease. 

Fig. 83 offers some suggestions in a pictorial way on 
how the capacity of the elbow may be maintained and 
distortion modified by increasing the hight of the duct 
at the intersection of duct and elbow. This will become 
necessary at each end of the elbow, that is, that modifica- 
tion shown in Fig. S3 would be also applied to the bottom 
of the duct in wall B. 



CHAPTER XXL 

When it is Required that a Round Pipe Should 
Join the Frustum of an Oblique Cone. 

An interesting and instructive problem when it is 
required that two elements of the conical form shall 
lie in planes which are at right angles to each 
other, and perpendicular to the plane of its base, 
with one element of the cylinder in one of said 
planes. 

While this problem is an unusual one, the author has 
occasionally noted instances where a fitting of this kind 
has been demanded. It is an instructive problem, as it 
involves the use of the supplementary plane and the in- 
troduction of intersecting surfaces to secure points in 
the line of contact between the cylinder and conical form. 

The intersecting surfaces, or cutting planes, the use 
of which will be pointed out in this demonstration, are 
important factors in the solution of many problems. They 
are many times employed in examples which are so simple 
that an understanding of their use is of small importance. 
On the other hand, the more complex examples demand 
an understanding of their use and importance. It has 
been here presumed that the patterns are required for a 
fitting as described above, and shown in a pictorial way 
in Fig. 86. 

The Plan and Elevation. 

A plan is the first step in developing the patterns for 
thisJdnd of fitting. We may therefore draw two lines at 

172 



CYLINDER AND CONii 



173 



right angles to each other, as / L and L D, Fig. 87, and 
look upon said lines as the plans of planes within which 
two elements of the conical form are situated. The plan 
of each end of the conical form will be secured when 
circles are drawn at the required diameters, and tangent 
to I L and L D. When the vertical hight of the conical 
form is known, an elevation of that part is secured in the 




Fig. 86. Cylinder Intersecting Frustum of a Scalene 
Cone. 

usual manner, and clearly shown by construction lines in 
Fig. 87. 

The first step in securing an elevation of the intersect- 
ing collar is to draw parallel lines as c J and 1 V , Fig. 87, 
at the required angle, and at a distance from each other 
equal to the diameter of that part. A partial plan of the 
collar is secured by drawing a line as C 6, which is 
parallel to / L, and at a distance from it equal to its 



174 TRIANGULATION 

diameter. We may now look upon a portion of the line 
I L as one element of the cylindrical form, i.e., that 
element which is farthest from the eye in elevation. 

Rectilinear elements of the conical form, or lines pre- 
sumed to be upon its surface, are located in the usual 
manner, after having presumed one point of division of 
each circle to be the exact point of tangency between said 
circles and the line which is a plan of the tangent plane, 
as shown at point 1 of each circle in plan. 

The elevations of these lines are secured in the usual 
manner, as is clearly shown by the vertical projectors. 
The next work is to locate points in the line of contact 
between the round collar and the conical form, as these 
points must be located before we can proceed with the 
patterns. These points are in reality one extremity of 
rectilinear elements of the cylindrical form which con- 
stitutes the round collar, and are best represented upon 
a plane which is parallel to said elements. This is pre- 
cisely what we have in the elevation shown. In proof of 
the foregoing, it will be recalled that all rectilinear ele- 
ments of a cylindrical form are parallel to the axis of that 
form, therefore parallel to each other. 

Upon examination it will be noted that a portion of 
the line 7" L, Fig. 87, is looked upon as a plan of one 
element of the round collar, therefore parallel to the 
vertical plane. 

Intersecting Surfaces or Cutting Planes. 

In all cases where what may be looked upon as one 
solid of revolution, such as a cylinder, cone or sphere, is 
joined, or made to penetrate as it were, another solid, a 
line is developed which is called the line of contact in the 
above, and also known as the line of penetration. This 
line of penetration is the important factor in examples 



CYLINDER AND CONE 



175 




176 



TRIANGULATION 



of this class, since we are hopelessly defeated unless it 
can be represented. 

Points in this line are located by the use of intersecting 
surfaces, i.e., by dividing each form by planes common to 
both. These surfaces, or planes, may be either perpen- 
dicular, parallel, or inclined to the planes of projection, 
or the axis of the form. However, to simplify the work 




Graphic Representation of 
the Cutting Plane. 

of finding the projection of said lines developed by said 
planes, it is desirable to fix upon a direction for the inter- 
secting surfaces which will give the most simple form, 
i.e., right lines or circles. The same problem may be 
simplified, or made more complex by the location of these 
planes. 

For example, a cylinder may be cut by a plane oblique 
to its axis, thus developing an elliptical form as shown in 
a pictorial way at the left of Fig. 88. Since the elliptical 
form is usually secured by locating a number of points, 



CYLINDER AXD CONE 



177 



it prolongs the work and invites confusion if there is a 
number in any one problem. If the cylinder be cut by a 
plane parallel to its axis, the form of section developed 
is a rectangle as shown at the right of Fig. 88. Here as 
will be noted, a true form of section may be represented 
by right lines. 

Fig. 89 shows in a pictorial way two intersecting 
cylinders which have been penetrated by a plane. As will 




Fig. 89. Graphic Representation of Two Inter- 
secting Cylinders Cut by a Plane. 

be noted, the intersecting surface is parallel to the axis 
of each cylinder, thereby cutting rectilinear elements 
from each. The intersections of said elements form 
points which are in the line of contact between the ver- 
tical and horizontal cylinders. This is precisely what 
our aim is to accomplish in the problem before us. 

From an analysis of the problem secured from the 
diagrams in Fig. 87 we conclude that intersecting sur- 
faces may be so located as to cut rectilinear elements from 
each form. The next work is then to find a plane upon 
which a representation of the object may be drawn, and 
said surfaces located. The position of this plane is found 



178 TRI ANGULATION 

when a line as A B, Fig. 87, is drawn perpendicular to 
lines 1 V and e J. That is, the line A B is now looked 
upon as the intersecting line between the vertical and an 
oblique plane. 

Relative Positions of Planes of Projection. 

Fig. 90 shows in a pictorial way the relative positions 
of the planes of projection. As, for example, we have 
before us the horizontal plane, above this is the vertical 
plane, and to the right is the oblique plane. Between the 
eye and the vertical plane there is suspended an object 
of which two faces are parallel to the vertical, with the 
remaining four faces at an angle to the horizontal, and 
perpendicular to the vertical plane. 

Since the oblique plane is parallel to the two smaller 
faces of the object, its representation upon that plane 
will be its true form when viewed in the direction in- 
dicated by the arrow A. The position of the diagram 
which represents the object upon the oblique plane is de- 
termined by the position of the object in space. This 
position is indicated by the plan and elevation. Thus the 
representation of the object will be at the same distance 
from the vertical plane as its plan, which is shown 
directly beneath it upon the horizontal plane. 

A Model May Prove of Service. 
If any difficulty is experienced in securing an under- 
standing of the planes as shown in a pictorial way at Fig. 
90, a crude model may be constructed, which will no 
doubt prove of considerable value in securing that under- 
standing. After having constructed a model of the 
planes, an object may be held in position as shown, and 
viewed as indicated by the arrows, remembering that in 
every case the point viewed is in a line perpendicular to 
the plane upon which it is represented. We may thus 



CYLINDER AND CON 



179 



secure a clear understanding" of die plan, elevation and 
representation of the object upon the oblique plane. Jt 
should also be remembered that Fig". 90 is simply a pic- 
torial representation. 

Since a portion oil L, Fig. S7, is a plan of one element 
of the round collar, the collar itself must be tangent to 
the vertical plane. Therefore if a circle be drawn upon 




Fig. 90. Graphic Representation of An Object 

In Space and Upon Vertical, Horizontal 

and Oblique Planes. 

the oblique plane in a position as shown, and of a 
diameter equal to the known diameter of the round collar, 
a representation of it is secured upon that plane. Each 
element of the conical form shown in plan and elevation 
may be represented upon the oblique plane as indicated 
by the oblique projectors, remembering that the upper 
and lower extremities of these lines will be of the same 
distance from the line A B as similarly designated points 
are from line / L in plan. The representations of recti- 



180 TRI ANGULATION 

linear elements of the cylindrical form upon the oblique 
plane are in points ab c d e etc., of the circle. The repre- 
sentations of said elements or lines in elevation are 
located as shown by the oblique projectors from said 
points. 

Since elements of the conical surface which were first 
located in plan do not cross or intersect all parts of the 
circle when represented upon the oblique plane, some 
additional lines must be introduced. In other words, 
when said elements are looked upon in the oblique pro- 
jection as the representations of planes which cut recti- 
linear elements from each form, there is not a sufficient 
number, therefore we locate lines as R R and S ,S\ These 
lines can be located by dividing the curved lines between 
points 6 and 7 of each end of the conical form shown in 
the oblique projection, into the same number of equal 
parts, after which those lines may be drawn through 
similar points of division. The positions of said lines 
upon the conical form may then be very closely approxi- 
mated, or they may be projected in the usual manner as 
shown, to the elevation and then to the plan, which will 
thus secure their exact location, providing this work is 
done accurately. 

It will be noted that the line 5 5* in the oblique pro- 
jection, when looked upon as the representation of a 
plane, cuts but one element from the cylinder, as shown 
at h, therefore becomes a tangent plane. 

To locate points which lie in the line of contact in 
elevation, we look upon points of the circle as a, b, c, d, 
etc., in the view upon the oblique plane, Fig. 87, as the 
representation of lines, the lower extremities of which 
intersect elements of the conical form. As for example, 
the lower extremity of line a is at its intersection with 
the conical element 1 1, and when shown in elevation is 



CYLINDER AND CONE 181 

at point E. Likewise lines shown at b and It in the 
oblique projection terminate upon coming into contact 
with the conical element 2 2, and shown in points / ; and 
X of the elevation. By similar reasoning" we locate the 
remaining points which lie in the line of contact, as 
G, H, /, K, 0, P, Q, T, U, V and Y. In locating the 
above points, some attention must be devoted to deter- 
mining upon which side said points are situated. This 
work will be simplified if we remember that those points 
shown upon the oblique plane which are farthest from 
the line A B, are nearest the eye in elevation, and con- 
versely, those points which are farthest from the eye in 
elevation are nearest the line A B upon the oblique plane. 

Pattern for the Round Collar. 

Having located points in the line of contact between 
the round collar and the conical form as shown in eleva- 
tion at points E, F, G, PI, etc., we are now in a position to 
secure the pattern for the round collar. 

Lines as b F, h P, i O, etc., of the elevation, are the 
elevations of rectilinear elements of the cylindrical sur- 
face, and are represented upon a plane which is parallel 
to said elements, therefore shown upon that plane in 
their true lengths. Points as a, b, c, d, c, f, etc., upon 
the oblique plane, furnish us with the distance said ele- 
ments are from each other upon the round collar. 

Therefore we may, in any convenient position, draw 
parallel right lines at distances from each other as found 
upon the circle in the oblique projection, as shown at the 
pattern for the round collar. One extremity of all recti- 
linear elements of the round collar terminates at its upper 
edge in one right line when said surface is developed 
into a plane. The lower extremities of said lines or ele- 
ments are in points E, F, G, H, etc., of the elevation. 



182 



TRIANGULATION 



Therefore we simply transfer lengths as found in eleva- 
tion to similarly designated lines upon the plane of de- 
velopment to locate points through which a line may be 
traced to represent the boundaries of the pattern for the 
round collar, as is clearly shown in Fig. 87. 




Fig. 91. Pattern for the Required Opening 
in the Conical Form. 



The one who has followed this work will have little 
difficulty in developing the pattern for the conical form, 
since this has been fully explained in the early chapters, 
therefore reference to that is here omitted. However, 
attention is directed to the fact that the only line which 
divided the plan of the conical form into equal parts is 
that shown at M L. 

If the pattern for the conical form was the sole con- 
sideration, its elevation would be best drawn upon a 
plane, the intersecting line of which is parallel to line 
M L. As this demonstration is for the purpose of illus- 
trating methods which may be pursued to secure the 
pattern for the intersecting collar and the required open- 
ing in the conical form, the diagrams are best drawn as 
shown at Fig. 87. 



CYLINDER AND CONE 183 

The following explanation has been written on the 
presumption that the reader is in a position to develop 
the pattern for the frustum of an oblique cone, and 
locate right lines upon said pattern, the plans of which 
are shown in Fig. 87. It will be recalled that in this 
demonstration a number of those lines have been looked 
upon as elements cut from that form by intersecting 
surfaces, which also cut rectilinear elements from the 
round collar, and the intersections of said elements lo- 
cated points in the line of contact between the round 
collar and the conical form. 

Locating Points In Required Opening. 

To locate points in the outline of the required opening, 
we must determine the exact points along the conical 
elements at which the cylindrical elements intersect 
them. In this demonstration it is presumed that lines 
as shown in plan and elevation have previously been 
located upon the pattern, and that our purpose is to 
locate points along said lines, which are in reality in the 
line of contact. 

Upon turning attention to Fig. 87 w r e note that tri- 
angles have been drawn whose longest sides are equal 
in length to those conical elements which are inter- 
sected by elements of the round collar. The method of 
drawing these triangles will no doubt be apparent, since 
the base lines of all are in the line N W , which is in 
reality a continuation of the base line of the fitting in 
elevation, and that the vertical hight of all is equal to the 
vertical hight of the conical form. Each triangle is 
designated by a number at the top. The true length of 
any one line is then found by reference to the character 
at the top of the triangles. As for example, the true 
length of line 3 3, shown in plan and elevation, is found 



184 TRIANGULATION 

in the hypothenuse of the triangle marked 3 at the top, 
and so on for all lines shown. 

Points of Contact in Elevation. 

Since we have in elevation the elevations of a number 
of points of contact between the round collar and the 
conical form, we may project said points parallel to the 
base of the fitting, and locate them upon corresponding 
lines in the diagram of triangles as shown at Fig. 87. 
For example, we may select the conical element 3 3, 
when it is noted that said element contains two points 
of contact as G and Y. Horizontal lines are drawn 
from said points to intersect the oblique line 3 in the 
diagram of triangles. These points of intersection are 
then the exact point of contact along the conical element 
3 3, and in their correct location as regards the extremi- 
ties of that element. 

Or we may select the conical element R R, when we 
find points of contact as O and ; these points may then 
be projected to the oblique line marked R in the diagram 
of triangles, to locate points O and O in their correct 
positions along the conical element R. R. By continuing 
this work for each point shown, in a manner as explained 
above, we are enabled to locate all points shown in eleva- 
tion. Since in this example the conical elements R R, 
S S and 6 6 are found to be of equal lengths, all are rep- 
resented in one line in the diagram of triangles. 

Having now before us the exact location of points of 
contact along the rectilinear elements of the conical 
form, we may transfer them to our pattern, presuming 
said pattern to have been developed as shown at Fig. 91. 
That is, we shall presume that the pattern has been de- 
veloped, and that lines as shown in plan and elevation at 
11,2 2, 3 3, etc., to have been located thereon. To 



CYLINDHK AND CONE 185 

locate these points along said lines is but to transfer 
distances as found in the diagram of triangles to similar 
lines of the pattern, taking all distances in the diagram 
of triangles from the line N IV. 

For example, we set our compasses to a distance equal 
to the distance from the line N W to point E, on the 
oblique line 1 of the diagram of triangles, and mark a 
similar distance upon element 1 1 of the pattern from 
its base, thereby locating point E as shown. In a similar 
manner we locate points F and X in the line 2 2 of the 
pattern, and so on for all points shown in the elevation. 
After which a line is traced through said points to com- 
plete the outline of the required opening, as shown, 
Fig. 91. 

The lengths of the lines which represent the opening 
in the conical form should now measure approximately 
the same as the lower edge of the pattern for the round 
collar, and if this proves to be the case, it is a fair indi- 
cation that our work is correct. 



CHAPTER XXII. 



A Branched Fitting Commonly Known as 
"Breeches." 



In some branches of sheet metal work, there is a con- 
stant demand for the branched fitting. As an intro- 
ductory problem to satisfy this demand, we shall here 




Fig. 92. Photographic View of the Fitting. 

presume that the pattern is required for a fitting as 
shown in a pictorial way at Fig. 92. Said fitting is de- 
signed to make connection between a trunk line and two 
smaller branches. The axes of all to be in the same 

186 



Branched Fitting 187 

plane, with the smaller pipes radiating from, or con- 
verging to the trunk line at an angle of 45 degrees. 

This is a problem wherein the necessary diagrams 
may be curtailed to a considerable extent in developing 
the pattern. However, to place before the student the 
reason for, and the use of these curtailed diagrams, we 
shall first consider a complete plan and elevation. 

A Complete Plan and Elevation. 

Having before us the required measurements, we may 
first draw a horizontal line as A B, Fig. 93, whose length 
is equal to the diameter of the large pipe. From the 
center point of this line as at C, erect a perpendicular as 
C D. Set off from C along line C D, a distance equal 
to the required length of the fitting as at E. Through 
the point E draw a horizontal line as F G. Upon line 
F G, and at each side of point E, set off one-half the re- 
quired distance between the small collars as at points 
F and G. From points F and G draw lines as F II and 
G I , which are at an angle of 45 degrees to line C D. 
Set off distances along said lines equal to the diameter 
of small collars, as shown at II and /. Draw lines as 
H A and I B to complete a view which is in this instance, 
looked upon as an elevation, or a section of the fitting 
taken upon line K M. 

When said diagram is looked upon as an elevation, a 
view of the fitting upon the horizontal plane is secured 
by first drawing a circle whose diameter is equal to the 
length of line A B, and in a position as shown, i.e., its 
center is in some point along the line C D produced. 

Lines as H F and G I are looked upon as the edge 
view of circles whose diameters are equal to the diameter 
of the small pipes. As will be noted, said circles are per- 



188 TRIANGULATION 

pendicular to the vertical plane, and at an angle to the 
horizontal, thus the representations of said circles upon 
the horizontal plane will be elliptical. To draw these 
forms in their correct relative positions, we draw semi- 
circles as shown, which are in reality semi-profiles of 
the round collars. Divide said semi-profiles into a con- 
venient number of parts as shown, and project these 
points of division to lines H F and G J, as also shown. 
From points thus located along lines H F and G J, we 
drop vertical projectors to the horizontal plane. 

From any convenient point along line N M, Fig. 93, 
we may draw a semi-circle whose diameter is equal to 
the diameters of the round collars, and divide said semi- 
circle into the same number of equal parts as the semi- 
profiles have been divided into. From said points of 
division we draw lines parallel to line I L as shown. 
Then will points secured in the intersections of these 
lines with the vertical projectors, be points in the plans 
of the openings to which the round collars are to be con- 
nected. As for example, if we look upon point a in the 
elevation as the end of a line which is perpendicular to 
the vertical plane, and whose length is equal to the length 
of line b b shown in the semi-circle M, the broken line d d 
in plan becomes a plan of that line. Since the positions 
of all other points which must be located in plan are 
secured by similar work and reasoning, the student 
should have little difficulty in comprehending, or draw- 
ing diagrams as shown at Fig. 93. 

With the plan as shown at Fig. 93 before us, we note 
that said diagram is capable of being divided into four 
equal parts, i.e., by lines K M and C N produced. Thus 
we conclude that diagrams may be drawn to represent 
one-quarter, and measurements thus obtained duplicated 
for the three remaining parts. 



BRANCHED FITTING 



189 



From an analysis of the fitting derived from its plan 
and elevation, Fig". 93, we also conclude that there arc 
two portions which closely resemble the conical form, 
and between these there are two flat triangular surfaces. 




Fig. 93. The Branched Fitting Represented in Plan and 
Elevation. 

The upper portion, or that to which the upper half of 
each collar is to be connected, is a form which when cut 
at the required angle, supplies a semi-circle as its section. 



190 TRIANGULATION 

Since the lengths of lines presumed to be upon one- 
quarter of the fitting may be duplicated for the remain- 
ing three-quarters, we may, when the pattern is de- 
veloped, curtail our diagrams as is shown in Fig. 94, i.e., 
it is only necessary to represent but one-quarter of the 
object in plan and elevation. 

A Curtailed Plan and Elevation. 

To draw a plan and elevation as shown at Fig. 94, 
we first draw the quarter circle in plan, to a diameter 
equal to the required diameter of the large collar. 
Through the point from which the quarter circle was 
drawn as at A, we draw a perpendicular line, and set off 
a length equal to the required length of the fitting, as 1 a. 
From point a, draw the horizontal line a e, locating point 
^ at a distance from a equal to one-half the required dis- 
tance between the round collars. From a point e, draw a 
line at the required angle to 1 a, as c 5. Locate the point 
5 at a distance from e equal to the required diameter of 
the small collar. From the center point of line e 5, draw 
a semi-circle whose diameter is equal to the length of line 
c 5 as shown. Divide said semi-circle into a number of 
equal parts, and project said points of division to the 
line c 5, as also shown at 2 3 4 and bed. 

From some point along the line 5 A", draw a quarter 
circle whose diameter is equal to the required diameter 
of the small collars as at Y, and divide this arc into the 
same number of parts as was a similar arc shown in 
the semi-profile, as at points 2 3 and 4. Draw indefinite 
horizontal lines through these points to intersect lines 
dropped from points 12 3 4 and 5 on the line 5 e, then 
will these intersections be points in the line which is a 
plan of the fitting on line 1 5 of the elevation. 



BRANCHED FITTING 



191 



Divide the quarter circle in plan into the same num- 
ber of parts as the arc 1 5 of the semi-profile has been 
divided into. Project said points of division to a hori- 
zontal line drawn through the lower extremity of line 
A 1 , thus locating points as 12 3 4 and 5 at the base of 
the fitting in elevation. As will be noted, we have thus 




Fig. 94. Diagrams from Which the Pattern May Be Secured. 

located lines in plan and elevation which may be pre- 
sumed to be upon the surface of the fitting, and which 
we shall use to develop the pattern. 

The plan and elevation before us supply the distances 
said lines are from each other; however, their lengths 
must be determined, therefore we construct a diagram 
of triangles as shown. 



192 TRIANGULATION 

The method of constructing the diagram of triangles 
is substantially the same as with all examples in this 
branch of pattern development, and to those who have 
followed this work, it is but a simple operation. As a 
matter of fact it is folly for one to attempt the solution 
of a problem of this nature without having first acquired 
some understanding of the more simple examples. 

As will be noted upon examination of Fig. 94, the 
elevation supplies the perpendicular hight for all tri- 
angles. For example, the triangle which must be con- 
structed to secure the length of line 1 A on line 1 A X 
of the plan, has a base equal to the length of line 1 A 
on the line 1 A X, and a perpendicular equal to the length 
of the vertical line A 1 of the elevation. 

The base of a triangle from which we may secure the 
true length of line 1 1 shown in plan and elevation, is 
equal to the length of line 1 1 of the plan, with a per- 
pendicular equal to the vertical distance between the 
extremities of that line shown in elevation, and so on 
for all lines presumed to be upon the surface of the 
fitting. 

It must be remembered that the rectilinear elements 
as 1 1, 2 2, 3 3, etc., are not sufficient to develop the pat- 
tern, therefore additional lines must be introduced, as 
shown in broken lines 1 2, 2 3, 3 4, and 4 5, and whose 
true lengths are also shown in the diagram of triangles. 

To Develop the Pattern for the Irregular Portion 

Having determined the true lengths of lines presumed 
to be upon the irregular portion of the fitting, and shown 
in plan and elevation, we may proceed to develop the 
semi-pattern for that part, when our line of reasoning 
may run somewhat as follows : Since we are to develop 



BRANCH 



ITTING 



193 



a half pattern from the diagrams before us, we must 
duplicate practically all measurements found in those 
diagrams. Therefore we draw in any convenient posi- 
tion, a line, some portion of which is the line 1 A, as 
shown at the vertical line 1 A, Fig*. 95. Having located 
a point as 1 at the base of the pattern to our satisfac- 
tion, we set off along that line a distance equal to the 




Fig. 95. Semi-Pattern for the Branched Fitting Shown at 
Fig. 92. 

true length of line 1 A found in the diagram of triangles, 
as shown at A of the pattern. This is a line which 
divides the flat triangular surface of the fitting into two 
equal parts. 

Through point A of the pattern, we draw a line per- 
pendicular to the first, making it of a length each side of 
A equal to the length of the horizontal line 1 A of the 
elevation, as shown at 1 A 1 of the pattern, Fig. 95. To 
complete the boundaries of the flat triangular surface, 
we draw the lines 1 1 as shown. 



194 TRIANGULAT10N 

The distance between lines 1 and 2 at the base of the 
fitting is found in the first division of the quarter circle 
in plan. Using* this distance as radius, and with the 
point 1 at the base of the pattern as center, we draw two 
small arcs as shown. With compasses set to a span 
equal to the distance between points 1 and 2 of the semi- 
profile, we use the points 1 at the top of the pattern as 
centers and draw arcs, as also shown. With the length 
of line 1 2 found in the diagram of triangles, and with 
point 1 at the base of the pattern as center, describe small 
arcs cutting the first at the top of the pattern, as shown 
in points 2. Then will point 2 be the upper extremity 
of the rectilinear element 2 2. With the point 2 at the 
top of the pattern as center, and with the length of line 
2 2 found in the diagram of triangles as radius, we draw 
arcs cutting the first at the base of the pattern, thereby 
locating the lower extremity of the rectilinear element 
2 2 in its correct relative position. 

To complete the pattern for the irregular or lower 
portion of the fitting as shown, is but a repetition of the 
work as explained above, using the length of each line 
shown in the diagram of triangles in rotation to locate 
said lines in their correct relative positions, remember- 
ing that the true distances between the lower extremities 
of said lines are secured from distances points of division 
are from each other in the quarter circle in plan which 
represents the large collar, and that the true distances 
between the upper extremities of said lines are secured 
from the semi-profile. 

Fig. 96 will no doubt be of service in securing an un- 
derstanding of this, since that Fig. shows in a pictorial 
way the semi-pattern formed to its required shape, with 
said lines upon its surface. 



BRANCHED FITTING 



195 



On the Portion of the Fitting Which May Be 
Looked Upon as a Parallel Form. 

The portion of the fitting shown in Fig. 94 above the 
horizontal line 1 A, is a parallel form whose cross-sec- 
tion or profile will show a semi-ellipse, or in this instance, 
i.e., in Fig. 94, where it is presumed that one-quarter 
of the fitting only is represented, said section will then 
be a quarter ellipse as shown at section on line a A. 
Therefore to develop this portion of the pattern, we may 




Fig. 96. Pictorial Representation of the Semi- 
Pattern When Bent to Its Required Form. 

look upon the horizontal line 1 A and lines above it, as 
b c d and e, as rectilinear elements of that surface, and 
in this instance, said elements are in their true lengths. 
If then we can determine the distance these elements are 
from each other, the development of that surface is but 
a simple matter. As will be noted, the section on line a A 
shows in points 1 b c d and e, those distances. This 
section is drawn by projecting indefinite horizontal lines 
from points 1 b c d and c as shown, and in any conven- 



196 TRIANGULATION 

ient position erecting a perpendicular as E e. From the 
line E e, set off distances on the several lines as shown. 
As for example, the distance from line E e to d is that 
found between points d and d of the semi-profile, and 
so on for each line represented. 

Presuming an understanding of this is secured, we 
may now complete the semi-pattern. That is, we draw 
lines parallel to line 1 A 1 of the pattern at a distance 
from each other equal to those distances found between 
similarly designated points in the section a A, and locate 
points upon said line each side of line e A of the pattern, 
at distances as found in the elevation, i.e., the length of 
the horizontal line shown in elevation whose left hand 
extremity is at point b, is set off each side of the pattern 
on the horizontal line which intersects point b. 

By applying similar methods to the remaining lines 
shown in elevation, or those whose left hand extremities 
are in points c d and e, we are enabled to complete the 
semi-pattern as shown. 

Some Variation May at Times Be Desirable. 

It is by no means necessary that the pattern be de- 
veloped precisely as here shown. In other words, we 
may if we wish, select other positions for the seams, or 
we may introduce more seams. For example, we may 
make the main body of the fitting in one piece, and that 
portion between the collars at the top as a separate piece, 
with seams on each side on a line as 1 A 1 of the pattern, 
or, we can if we wish, cut the whole from one piece. 

Since lines which are presumed to be upon the sur- 
face of the fitting, and represented in plan and elevation 
are presumed to be right lines, we should at all times use 
care in their location. That is, their positions should be 
so taken as to allow said lines to be as nearly straight as 



BRANCHED FITTING 197 

possible if placed upon the surface of the object. In 
this example, slightly more accuracy may be obtained by 
presuming- the broken lines to connect points as 5 of the 
base to 4 of the top, and so on. However, some slight 
inaccuracy will usually appear in examples of this 
nature. On the other hand, we should not be too quick 
in assuming- an error. Be sure your metal has been 
made to assume its intended form before judgment is 
passed. 



CHAPTER XXIII. 

A Simple Two Pronged Fitting. 

Fig. 97 illustrates a two pronged fitting of the most 
simple order. No doubt a form as here shown will re- 
ceive some criticism, which may in some instances be 
justifiable. However, since it is a problem containing 




Fig. 97. Photographic View of the Fitting. 

principles which may be employed in securing the pat- 
terns for the more popular forms of branched fittings, it 
is here introduced in an endeavor to convey to the student 
that understanding necessary to enable him to develop 
the patterns for the more complicated forms. 

198 



TWO PRONGED FITTING 



199 



As illustrated at Fig. 97, the axes of all collars arc in 
one plane, or as we hear it in the shop, il is "on center." 
A change of the plan places all collars tangent to one 
plane, or what is commonly known as "flat on one side." 




Fig. 98. The Plan and Elevation of a Fitting as 
Shown at Fig. 97. 

For the moment we shall presume that the pattern is re- 
quired for a fitting as illustrated at Fig. 97, and as the 
work progresses, endeavor to explain the methods which 
may be pursued to secure the pattern when it is required 
that the fitting shall be "flat on one side." 



Diagrams to Represent the Object. 

Simple diagrams will in this instance represent the 
object in plan and elevation, as shown at Fig. 98. To 



200 TRIANGULATION 

draw these diagrams, we may first draw a circle whose 
diameter is that of the main stem, as shown at F G. 
From the center of said circle as at A, draw an indefinite 
vertical line as A B. Locate a point along- this line as C, 
which shall be in the base line of the fitting in elevation. 
Draw a horizontal line through point C as D C F, and 
project points F and G of the large circle in plan to the 
line D C F, as at points D and E. Set off from C along 
the line C B, a distance approximately equal to the length 
of line F G as shown at H. Draw lines as F H J and 
D H M. At a reasonable distance above point H on 
lines D H and F H, locate points as / and M. Through 
points / and M draw horizontal lines as shown, and set 
off distances equal to the diameters of the small collars 
as shown at K and N. Draw lines as K D and N E to 
complete the diagram here looked upon as an elevation. 
Perpendicular lines dropped from points K J M and N 
to intersect the line O Q locate points through which the 
circles are drawn to represent the small collars in plan. 

The Length of the Fitting at the Intersection 
of Its Prongs. 

The distance set off along line C B, as C H is by no 
means arbitrary, since as will be noted, it represents the 
length of the fitting at the intersection of its prongs. 
On the other hand, it has been found that a fitting of this 
class assumes a somewhat more symmetrical form when 
this length is made approximately equal to the diameter 
of the main stem. Make it more or less if conditions de- 
mand it, although a great departure from this rule will 
be found to distort the fitting. 



two pronged fitting 201 

The Form of the Object at the Intersection of 

Its Prongs. 

From an analysis of the fitting represented in plan and 
elevation at Fig. 98, we conclude that a portion of the 
elevation as shown at K J D E may be looked upon as 
the elevation of the frustum of a scalene cone, and that 
the two circles directly beneath it are the plans of the 
upper and lower extremities. If the conical form be 
cut away to the right of line C H, the remaining portion 
to the left of that line will then supply one prong of the 
fitting, and since the conical form is cut away through 
the center of its base, it may be duplicated for the op- 
posite prong. As will be noted, this mode of procedure 
allows the conical form to establish the form of the fit- 
ting at the intersection of its prongs. This, the author 
believes to be the most satisfactory course to pursue, 
since distortion at this point will thus be eliminated, or 
at least reduced. 

It is not to be understood that a form cannot be pre- 
established for the object at the intersection of its 
branches and results secured, providing one is compet- 
ent to establish a suitable form. Where the axes of all 
collars are in one plane as here represented, this is not a 
particularly difficult task. On the other hand, if the 
collars are required to be tangent to one plane, this work 
becomes more complex. 

Diagrams from Which a Pattern May Be Secured. 

When the pattern is required for a branch of given 
dimensions, we may secure the patterns for the frustum 
of a scalene cone, with diameters of its ends equal to 
those of the required fitting, and cut away a portion as 
above described. Upon examination of Fig. 98 we note 



202 TRIANGULATION 

that the line O Q divides the plan into equal parts, and 
that the line C A produced, also divides that diagram 
into equal parts, therefore we curtail our diagrams as 
shown at Fig 99. Here as will he noted, there is shown 
the semi-plan and elevation of the frustum of a scalene 
cone. To develop the pattern for this, and locate lines 
as shown in plan and elevation is but a simple operation, 
and -has been fully explained in foregoing chapters. 
Therefore to avoid undue repetition, it is here presumed 
that the student is in a position to develop the pattern for 
the frustum of a scalene cone from diagrams as shown 
at Fig. 99. The complete semi-pattern is here shown for 
the conical form, together with full lines presumed to be 
upon its surface, and shown in plan and elevation as 1 1, 
2 2, 3 3, etc. 

It should be remembered that this pattern was not de- 
veloped without the use of the indirect or broken lines, 
although here omitted in an endeavor to avoid all un- 
necessary confusion. If the student fails to secure an 
understanding of the methods pursued to secure the 
pattern for the frustum of the oblique or scalene cone as 
here shown, some attention given to chapter 6 should 
clear this portion of the problem. 

To Locate the Line upon the Pattern Which is 
at the Junction of the Prongs. 

We may look upon the line A 5 of the elevation, Fig. 
99, as the edge view of a plane which intersects or cuts 
away a portion of the conical form, and the line 5 5 as 
its plan. Said plane thus intersects the conical element 
5 5 at the base of the object, and cuts elements 4 4,3 3, 
2 2, and 1 1, as shown in points ABC and D. We may 
then construct a diagram of triangles in a position as 



TWO PRO N GET) FITTING 



2C3 




204 



TRIANGULATION 



shown, which will furnish the true lengths of the conical 
elements 11,2 2, 3 3 and 4 4, and draw horizontal lines 
from points ABC and D of the elevation, to intersect 
similar elements in the diagram of triangles, as shown. 
This will, as may be noted, supply the exact positions of 
points along said elements at which the cutting planes 
intersect them. To locate said points upon the pattern, 



/ 



H ; ; in i i 

ll j 12 173 I 




L 



Fig. 100. Showing a Plan to Be Substituted When It 
is Required That the Fitting Be "Flat on One Side." 

we transfer distances as found in the diagram of tri- 
angles to similarly designated lines upon the pattern, 
thereby locating points as A B C and D of the semi- 
pattern. 

A line traced through said points is the line upon 
which the conical form should be cut away, and which 
is at the junction of the prongs when the pattern is 
duplicated, and bent into its required form. 

It should be remembered that the pattern shown at 
Fig. 99 is simply the pattern for one-half of one prong, 
and must be duplicated for the other half. In other 
words, the body of the fitting will require four pieces of 
the pattern as here shown, 



TWO PRONGED FITTING 205 

To Secure the Pattern eor the Fitting When It 
Js Required to be ''Flat on One Side." 

When it is required that the fitting shall be flat on 
one side, the plan may be drawn to conform to this de- 
mand by drawing the circles which represent the ends of 
the object, tangent to one line. Fig. 100 is a diagram 
which fulfils this requirement in so far as one prong is 
concerned. If said diagram be substituted for the plan 
shown in Fig. 99, we may proceed in the same manner 
as has been explained, although it must be remembered 
that since this diagram cannot be divided into equal 
parts by lines which are parallel or perpendicular to / L, 
the true lengths of all lines must be secured, i. e., those 
lines presumed to be upon the surface of the object, and 
connecting points at the base and top. 

Points of division have been so taken in Fig. 100 that 
the elevations of said lines remain the same. As for ex- 
ample, the line 2 2 in elevation, Fig. 99 is the elevation of 
a line, the horizontal projection of which is 2 2 of the 
plan, and so on. When Fig. 100 is substituted for the 
plan shown in Fig. 99, we note that a portion of the lines 
in elevation, then become elevations of not only those 
lines nearest the eye, but of similar lines which arc 
farthest from the eye. As for example, the line 2 2 in 
elevation is not only the elevation of a line the plan of 
which is 2 2, but also represents a line whose plan is 
15 15, Fi t g. 100, and so on. Plowever, since lines shown 
in plan, Fig. 100, are of unequal lengths, the true lengths 
of all must be determined separately, thereby creating 
additional work and lines in the diagram of triangles. 
In addition to this, the whole pattern for one prong must 
be developed, and will be found to assume a somewhat 
different appearance from that shown in Fig. 98. 



206 TRIANGULATION 

After the pattern has been duplicated to form the op- 
posite prong", care must be used in forming, i.e., those 
parts must be formed in opposite directions to allow said 
parts to occupy correct positions when the fitting is as- 
sembled. 



CHAPTER XXIV. 

A Two Pronged Fitting Whose Prongs Are 
Unequal. 

The chief difficulty which is usually encountered when 
an attempt is made to develop the pattern for an unequal 
branched fitting as illustrated at Fig. 101 is to establish 




Fig. 101. Photographic View of the Fitting. 

a suitable form at the junction of its prongs. Therefore 
our first work will be to discuss methods which may be 
pursued to establish that form. 

Upon comparing Fig. 97 in Chapter 23 and 101, it 
will be noted that one prong of Fig. 101 is a duplicate of 

207 



208 TRIANGULATION 

the left hand prong of Fig. 97 : thus if we can determine 
the true form of a fitting as shown at Fig. 97 at the junc- 
tion of its prongs, we have established a form for that 
part of the fitting as shown at Fig. 101. 

To Establish the Form of the Fitting at the 
Junction of Its Prongs. 

When following a course as here suggested, we pre- 
sume one prong of the fitting to be a portion of the 
frustum of an oblique cone, and determine the true form 
upon the line which represents the junction of the prongs 
in elevation, which may be accomplished as shown at 
Fig. 102. 

Upon examination of Fig. 102, it will be noted that 
there is shown the plan and elevation of the frustum of 
an oblique cone, a portion of which will supply one prong 
of the required fitting. Fig. 102 also shows a number of 
lines which connect points of the top and base. The 
elevations of said lines are located by projecting the 
points of division of the circles to lines parallel to / L 
which represent the top and base of the object in eleva- 
tion. 

The line 5 13 in plan Fig. 102, is a plan of a plane 
which cuts away a suitable portion of the conical form, 
since it passes through the center of the large circle, and 
as said line is perpendicular to / L, the line 5 D is an 
elevation of said plane. This plane thus cuts the conical 
elements or lines which connect points of the top and 
base in points E F G H J K and L, and intersects ele- 
ments at the base of the fitting in points 5 and 13. The 
distance above the base of the object at which said plane 
intersects or cuts these elements is shown at points A 
B C and D of the elevation. 

Thus we have definitely located in plan and elevation 



TWO PRONGED FITTING 



209 



a number of points upon the surface of the conical form 
which were created by the plane in penetrating that 
form. Since the position of a point in space may always 
be determined from its plan and elevation, we may pro- 




Fig. 102. Plan, Elevation, and True Form of Section. 



ceed to locate those points in their correct relative posi- 
tions by erecting perpendicular lines at distances from 
each other as found between points along the line 5 13 
of the plan as shown at the true form. Said lines are 
now intersected by lines projected from points D C B 
and A, thereby locating points as shown at 5 E F G H J 
K L and 13 of the true form on line D5. A line traced 



210 TRIANGULATION 

through said points will then supply the form of the 
fitting at the junction of its prongs. 

Other lines of reasoning may of course be applied to 
this operation to secure identical results, as for example, 
the point B in elevation may be looked upon as the end 
elevation of a line which is perpendicular to the vertical 
plane, the length of which is the distance between points 
F and K of the plan, and so on for all lines shown. 

Having established the form of the fitting at the junc- 
tion of its prongs, we may proceed to develop the pat- 
tern for the right hand prong as shown at Fig. 103. 
Here as will be noted, an elevation may be drawn to sat- 
isfy the demand. In this instance it has been presumed 
that the lines A 5, 5 1,11 and 1 9 supply an 
elevation for a suitable form. With the elevation 
drawn in outline, our next work is to secure a plan of 
the object, together with lines presumed to be upon 
its surface. 

It will be noted that Fig. 103 shows a semi-plan of 
the frustum of an oblique cone, which was used as one 
prong of the fitting discussed in the last chapter. In 
this manner the lower extremities of lines presumed to 
be upon the surface of the right hand prong may be con- 
veniently located as shown. As the length of line 1 9 
represents the diameter of the small collar, we draw a 
semi-circle as at F, which then becomes a profile of that 
portion of the object. Thus we have a semi-circle which 
is perpendicular to the vertical plane and at an angle to 
the horizontal, a plan of which will be semi-elliptical as 
shown. To secure this semi-ellipse divide the semi-circle 
E into the same number of equal parts as was the semi- 
circle which represents the base of the conical form. 
Said points of division of the semi-circle E are projected 
to the line 1 9 as shown in points 2 3 4, etc. From these 



TWO PRONGED FITTING 211 

points of intersection along the line 1 9 vertical lines are 
dropped to the horizontal plane and made of a length 
below the line / L as found in the semi-circle E. In this 
manner points are located in plan as 2 3 4, etc. A line 
traced through said points then supplies the semi-ellipse 
which is a plan of the semi-circle whose edge elevation is 
the line 1 9. 

The construction lines shown in plan and elevation 
Fig. 103 clearly show the method employed to locate 
lines which we shall place upon the plane of development 
in their true lengths and positions to secure points 
through which the outline of the pattern is drawn. As 
for example, those lines whose upper extremities are in 
points 6 7 8 and 9, connect points as A B C and D, which 
are the lower extremities of elements of the conical form 
when said form has been cut away, and are in reality 
points A B C and D of the true form on line A 5. The 
true form on line A 5, as shown at Fig 103, has been 
established in the same manner as shown in Fig. 102; 
however, in this instance, only one-half is shown, or that 
portion represented in plan, since the fitting is that com- 
monly known as "on center." Those lines whose upper 
extremities are in points 12 3 4 and 5 connect points in 
the lines which were originally presumed to be in the 
base of the conical form. Broken or dotted lines must 
be employed as in practically all examples of triangula- 
tion. 

Triangles. 

To determine the true length of lines presumed to be 
upon the surface of the object and now located in plan 
and elevation, triangles are constructed in the usual 
manner. The length of base for each triangle is found 
in the plan, and the perpendiculars are secured from the 



212 



TRIANGULATION 




b. 



Oh 



to 






TWO PROXGKlJ FITTING 



2i; 



elevation, as is clearly shown by the horizontal 

projectors. 

The Pattern. 

Having' before us in the diagram of triangles Fig. 103, 
the true lengths of lines which are presumed to be upon 
the surface of the object, and previously located in plan 
and elevation, we may proceed to develop the pattern 




Fig. 104. Semi-Pattern for One Prong of 
a Fitting as Shown at Fig. 101. 



somewhat as follows : Tn any convenient position upon 
the plane of development we draw a line whose length is 
equal to the length of line 1 1 of the elevation, or as 
found in the diagram of triangles and shown at 1 1 of 
the semi-pattern, Fig. 104. The distance between lines 
1 1 and 2 2 at their extremities is the distance between 
points 1 and 2 of the large circle in plan for their lower 
extremities, and one space of the semi-circle E at their 
upper extremities. Therefore we set our compasses to a 



214 TRIANGULATION 

span equal to the distance from 1 to 2 of the large circle 
in plan, and from the lower extremity of line 1 1 of the 
pattern, describe a small arc as shown at 2. With com- 
passes set to a span equal to one space of the semi-circle 
E, place one point at the upper extremity of line 1 1 of 
the pattern a rid describe an arc as also shown. The 
upper and lower extremities of line 2 2 must then lie in 
some points of these arcs. 

We note that the broken line 1 2 shown in plan and 
elevation connects point 1 at the base with point 2 at the 
top, and point 2 at the top is in some point of the small 
arc just drawn. Therefore we set our compasses to a 
span equal to the true length of line 1 2 found in the dia- 
gram of triangles, and describe a second small arc whose 
center is point 1 at the base of the pattern. In this man- 
ner Ave definitely locate the upper extremity of not only 
line 1 2, but line 2 2 as well. The lower extremity of 
line 2 2 is in the small arc at the base of the pattern, 
therefore we draw a second arc whose radius is equal to 
the true length of line 2 2, and whose center is point 2 
at the top of the pattern. The point of intersection be- 
tween those arcs as at point 2 at the base of the pattern 
must then be the lower extremity of line 2 2. 

By similar work and reasoning, we are enabled to lo- 
cate lines upon the plane of development as shown in 
Fig. 104. The pattern cutter should not lose sight of 
the fact that the distances as 5 D, D C, C B and B A 
shown at the pattern are in theory at least, secured from 
the true form on line A 5. 

Tn practise a more accurate course to pursue is to 
secure these measurements from the pattern for the left 
hand prong, which we may presume to have been first 
developed. This, as will be noted, not only increases our 
accuracy, but enables us to develop the pattern for each 



TWO PRONGED FITTING 



215 



prong without first finding the true form at the junction 
of the prongs. 

It may be here remarked that as has been previously 
stated, a form could have been established for the junc- 
tion of the prongs, and the pattern for each prong de- 




7 ; iV- 105. Plan, Elevation, and True Form of Section When It is Required 
That the Fitting Be "Flat on One Side." 



veloped in the same general manner as lias been ex- 
plained for the left hand prong". This is by no means a 
difficult operation, since the true form in this closely 
resembles a semi-ellipse whose major diameter is approx- 
imately double that of the minor. The author has noted 
very satisfactory results where this course has been 



216 TRIANGULATION 

pursued, although it is hardly to be recommended in 
every instance. 

When It Is Required That the Fitting Be "Off 
Center" or "Flat on One Side." 

When it is required that the fitting be "off center" or 
"flat on one side" the work of developing" the pattern is 
increased, since the object can no longer be divided into 
equal parts, consequently the surface of the whole object 
must be developed. In this, as with many other ex- 
amples, the principles involved are precisely the same; 
however, there is an increased number of lines to be dealt 
with, as was explained in the twenty-third chapter. The 
form of section, or the form at the junction of its prongs 
becomes far more difficult to approximate, and the 
methods here recommended are very likely to produce 
more satisfactory results than could be obtained by first 
establishing an arbitrary form for the junction of the 
prongs. 

Fig. 105 shows the true form of section of the conical 
form when said form has been presumed to be cut by 
a plane represented in plan and elevation by line 5 D. 
Here, as will be noted, the circles which represent the 
upper and lower extremities of the conical form in plan 
have been drawn tangent to the line / L, or a line which 
is parallel to it. This places one side of the fitting 
tangent to one plane, or "straight on one side." 

It may be here remarked that no absolute rule is in- 
tended to be laid down for the development of the pat- 
tern for the branched fitting. On the other hand, prin- 
ciples and methods are pointed out which have in past 
years been found to be of service to those whose work 
is to design and develop the patterns for various forms 
of branched fittings. 



CHAPTER XXV. 

On the Two Pronged Fitting When It Is Required 

That the Prongs Radiate at a Given 

Angle to the Main Stem. 

Attention will be here directed to methods which may 
be pursued to secure the patterns for a two pronged 
fitting whose prongs are required to radiate at a specified 




Fig. 106. Photographic View of the fitting. 

angle to the main stem. Fig. 106 illustrates a fitting 
which conforms to the above specification, and has been 
presumed to have been constructed by utilizing two 
prongs whose patterns were discussed in the twenty- 
fourth chapter. 

217 



218 



TRT ANGULATION 



This mode of procedure will produce very satisfactory 
results, although we are dependent upon our ability to 
establish a suitable form at the junction of the prongs. 
This can be accomplished by pursuing methods as ex- 
plained in the twenty-fourth chapter. However, that 
course is not to be recommended in every instance, there- 
fore methods which differ to some extent will be here 
discussed. 

If called upon for the pattern for a fitting as illus- 




Fii 



107. An Elevation, or a Section of a Two 
Pronscd Fitting. 



trated at Fig. 106, or one in which the prongs are re- 
quired to radiate at a specified angle, we may first draw 
an elevation as shown at Fig. 107, which is a simple 
diagram and in reality a section of the object. Upon 
examination of Fig. 107, we note that the line A B rep- 
resents the base of the fitting, or the edge view of a 
circle whose diameter is equal to the length of that line. 
In like manner, the lines D E are looked upon as repre- 
senting the diameter of the small collars, while the lines 
C E indicate the angle at which the branches are to 



TWO PRONGED FITTING 219 

radiate from the main stem. The line C F represents 
the length of the fitting at the junction of its prongs. 

Since Fig. 107 may be looked upon as an elevation of a 
fitting whose prongs are equal and "on center," the pat- 
tern can he developed for one-half and duplicated for 
the remaining portion as will he hereinafter discussed. 
Upon referring to Fig. 108 it will he noted that the dia- 
gram 1 9 E E 1 is a duplicate of one-half of Fig. 1 07, or 
an elevation of one prong of the fitting shown in eleva- 
tion or section at that Fig. Since the author is convinced 
that in the majority of cases it is far hetter to determine 
a form for the object at the junction of its prongs from 
some portion of it, than to establish an arbitrary one, it 
will be here shown how this may be accomplished if 
thought more satisfactory by the operator. From the 
above it is not to be inferred that an arbitrary form can- 
not be established and results secured by those who have 
given the subject some attention. 

Presuming we have before us an elevation of one 
prong of the fitting as shown at 1 9 E E 1, Fig. 108, we 
note that the line 1 9 represents the diameter of the 
small collar, and that the line 1 E represents one-half 
the diameter of the large collar. 

Since the base line of the object is here presumed to 
be in the line / L we may continue the line 9 E until it 
intersects the line /L as shown at 9 E 9. In this man- 
ner we secure a diagram which may be looked upon 
as an elevation of an object which has an oblong base 
and a round top. The major diameter of its base then 
becomes the length of line 1 9 E 9, with a minor diameter 
equal to that of the large collar at the base. A semi- 
plan of the base is then drawn as shown. A plan of the 
top is drawn as also shown, and was explained in the 
last chapter. 



220 



TRlANGULATiON 




TWO PRONGED FITTING 221 

Having* drawn the semi-ellipse in plan, which is a 
plan of the round collar at the top, we have located a 
number of points in said semi-ellipse which may be 
looked upon as points of division of the top. We may 
now divide the arcs in plan which represent the base of 
the object as shown, and draw lines as 2 2, 3 3,4 4, etc. 
In this manner we complete a semi-plan and elevation of 
an object, a portion of which will supply one prong of 
the required fitting when cut away as shown at line E 5 
of the plan, and line E E of the elevation. Upon a mo- 
ment's reflection, we are convinced that there are at 
least two courses open to us in developing the pattern. 
One is to develop the semi-pattern for the whole object 
as shown in elevation and cut away that portion to the 
right of line E E, as was explained in the twenty-second 
chapter. The second, which is here suggested and ex- 
plained, is to determine the true form of the object upon 
line E E, and utilize that as the true form of the branch 
at the junction of its prongs. 

The construction lines in Fig. 108 clearly show the 
method of locating lines in plan and elevation which are 
the plans and elevations of lines presumed to be upon the 
surface of the object. Presuming the object to be cut 
by a plane whose plan and elevation is line E E 5, said 
plane then cuts elements as shown in points A B C D and 
E. The plan then supplies in points 5 A B C D and E, 
distances from each other at which we may draw vertical 
lines in a convenient position as shown at the true form 
on line 5 E. The elevation supplies in points A B C D 
and E the distance above the base of the object at which 
these lines terminate, as at points A B C D and E of the 
true form on line 5 E, thereby establishing the true form 
of the object on line 5 E E, Fig. 108. 

Having now established the true form of one prong 



222 TRTANGULATION 

of the fitting at its extremities we may proceed to de- 
velop the semi-pattern by first constructing a diagram of 
triangles as is clearly shown in Fig. 108. As for ex- 
ample, the true length of line 4 4 shown in plan and 
elevation is found in the hypothenuse of a right angled 
triangle whose base is equal in length to line -/ 4 of the 
plan, and whose perpendicular is equal to the vertical 
distance the upper extremity of that line is above the 
line / L in elevation. This line of reasoning applied to 
all lines shown will enable one to construct his diagram 
of triangles as shown, or to determine the true length of 
any line he may select. 

Before proceeding with the pattern, it may be well to 
remind the student that in this example it is presumed 
that the fitting is to make connection between pipes whose 
axes are in one plane, or "on center." It will only be 
necessary to secure the pattern for that portion shown in 
plan at 1 E 5 5 1 , which may be duplicated to complete 
the fitting. 

The Pattern. 

Place upon the plane of development a line as 1 1, Fig. 
109, whose length is found in the diagram of triangles. 
The true distance between points 1 and 2 at the base of 
the fitting is found in plan, and the distance at the top 
of the fitting is one space of the semi-circle which repre- 
sents a semi-profile of that end. The true distance from 
1 at the base to 2 at the top is the length of line 1 2 
found in the diagram of triangles. Thus we are enabled 
to locate point 2 at the top in its correct relative position 
in the usual manner as shown. Since the work of lo- 
cating points 2 3 4 and 5 shown in the pattern has been 
frequently explained in foregoing chapters, a detailed 
description of that is here omitted. 

The true distance between points 5 and A at the junc- 



TWO PRONGED FITTING 



223 



tion of the prongs is found in the true form on line 5 E, 
thereby enabling us to locate line 5 A of the pattern, 
Fig! 109. 

The remaining lines as 6 B, 7 C, 8 D and 9 E are now 
located in the same general manner as has been ex- 
plained, thereby completing a semi-pattern as shown, 
which may be duplicated to complete the object. As 




Fig. 109. Semi-Pattern for a Two 
Pronged Fitting. 



will be noted, this process involves a flat triangular sur- 
face upon each side of the prongs as shown in elevation 
at A E 5. This, however, will be found to be of small 
importance in the finished fitting. 

Should the pattern cutter be called upon for a fitting 
of this class which is "off center," or with all collars 
tangent to one plane, some attention to foregoing chap- 
ters in conjunction with this, should place him in a posi- 
tion to develop its patterns without difficulty. 



CHAPTER XXVI. 

On a Fitting With any X^umber of Prongs. 

In foregoing chapters methods have been suggested 
which enable us to reduce the development of patterns 
for a two pronged fitting to a comparatively simple 
operation. Here attention will be directed to the devel- 
opment of patterns for the branched fitting of three or 




more prongs by utilizing the same form as was sug- 
gested in the last chapter. 

For example, a form whose elevation is shown at Fig. 
108, Chapter 25, was presumed to be cut away on line 
E E. With Fig. 108 before us it will be noted that the 
line E E is the elevation, and that the line 5 E is the plan 
of a plane which cut said form. This plane then passes 
through the center of the circle which is a plan of the 

224 



THE BRANCHED FITTING 225 

base of the fitting, thereby cutting said circle into two 
equal parts. If this form had been cut by a combination 
of planes whose plans included a sector of this circle equal 
to one-third of it, and properly placed, a portion of the 
original form could then have been utilized as one prong 
of a three pronged fitting. 

Again, had these cutting planes been located in such 
positions as to allow their plans to include a sector of the 
circle which is a plan of the base of the fitting equal to 
/ ^ L 



kJ^ 


«T^-= 






\ -\ 

\ \ 

\ V 


X. 




J^ 




\ \ 


\r 






i I 
1 


G 


C**^- 






/ 7 
___/ / 

/ — ~7^ 



Pig. 110. The Plan of an Object, a Portion of Which 
May Be Utilized as One Prong of a Fitting of a Con- 
siderable Number of Prongs. 

one-fourth of it, then a portion of the original form could 
he utilized as one prong of a four pronged fitting, and 
so on for any reasonable number of prongs. 

That we may better understand the above, Fig. 110 
has been drawn. This, as will be noted, is a complete 
plan of an object whose semi-plan and elevation has been 
shown in Fig. 108, Chapter 25. Here it has been as- 
sumed that the circle C D B E is a plan of the main stem 
of the required fitting, and the ellipse is a plan of the 
small end of one prong. The lines A B and A C have 
been drawn at angles of 60 degrees to line A D, or 120 
degrees from each other, therefore the sector A C D B 
is one-third of the whole circle, and that portion of Fig. 



226 TRIANGULATION 

110 to the left of lines A B and A C may be looked 
upon as the plan of one prong of a three pronged fitting. 

The lines A F and A G have been drawn at an angle 
of 45 degrees to line A D, or 90 degrees from each other, 
thereby including a portion of the circle which repre- 
sents the main stem of the fitting equal to one-fourth of 
the whole. Thus it will be noted that portions of Fig. 
1 10 to the left of lines A F and A G may be looked upon 
as the plan of one prong of a four pronged fitting. By 
similar reasoning we may look upon that portion of Fig. 
110 to the left of lines A K and A IF as the plan of one 
prong of a six pronged fitting, although in this instance 
the original object should be made somewhat higher, 
since the lines A K and A FI approach the ellipse or 
round collar at the top too closely for satisfactory results. 

In this manner we assume those lines radiating from 
point A as A B, A K, etc., as the plans of planes which 
may be employed to cut the original object in such posi- 
tions as to allow the use of the remaining portions of 
the object to be used as one prong of a fitting with a 
considerable number of prongs. 

We will now proceed to develop the pattern for a four 
pronged fitting in accordance with the above analysis 
of the problem. However, since the line D E, Fig. 110, 
divides that diagram into equal parts, a semi-plan will 
fulfil every requirement and curtail our work to some 
extent. 

Ttte True Form of Section. 

Fig. 1 1 1 shows a semi-plan and an elevation similar 
to that shown in Fig. 108, Chapter 25. Since the line 
A G, Fig. 110, is looked upon as the plan of a plane 
which cuts elements of the original form, we have only 
to determine the exact points at which those elements 



THE BRANCHED FITTING 



227 




228 TRIANGULATION 

were intersected by said planes to establish the form 
of the required fitting at the intersection of its prongs. 
Points as A B C ' D E F and G produced by the intersec- 
tion of line A Y with those lines which are elements 
of the original form are the plans of those points. Ver- 
tical lines drawn from points A B C D, etc., of that view, 
then supply us with the exact distance these points are 
from the horizontal plane, i.e., the distance said points 
are above the line / L. 

Thus we may set off along the line / L distances as 
found along the line A Y, and erect vertical lines as 
shown at the true form of section A Y. Horizontal lines 
drawn from points A B C D, etc., of the elevation to in- 
tersect those lines then supply points through which the 
line may be traced to secure a true form of section as 
shown. 

Triangles. 

The method of drawing the necessary triangles which 
will supply the true lengths of lines presumed to be upon 
the surface of the object, and shown in plan and eleva- 
tion, differs in no essential respect from those methods 
previously explained. The length of the lines shown in 
plan is the base, and the difference in hight of the ex- 
tremities of that line shown in elevation, is the perpen- 
dicular. As for example, to determine the true length 
of the line 8 B shown in plan, we may draw horizontal 
lines from the extremities of line 8 B in elevation as 
shown, and at any convenient point draw a perpendicular 
whose length is equal to the distance these lines are from 
each other. Set off along the lower horizontal line from 
the above spoken of perpendicular a distance equal to the 
length of line 8 B found in the plan. This, as will be 
noted, supplies points between which a line may be drawn. 



THE BRANCHED FITTIXG 



229 



which is in reality the third side of a right angled tri- 
angle which furnishes the true length sought, and shown 
at 8 B of the diagram of triangles. Since all other re- 
quired true lengths will he found in the same general 
manner, a detailed explanation would seem needless 
repetition. 

It will, no doubt, demand some attention to determine 
which lines would he intersected by the plane A Y. 



xs 




Fig. 112. Semi-Pattern for One Prong of a 
Four Pronged Fitting. 

However, the construction lines shown in plan and ele- 
vation clearlv show them. 



The Semi-Pattern for One Prong. 

Having now in mind the principles and methods 
which must be pursued to secure the true length of lines 
presumed to be upon the surface of the object, we may 
proceed to draw upon the plane of development those 
lines in their correct relative positions as shown at Fig. 
112. Here, as will be noted, the line 1 1 is drawn whose 



230 TRIANGULATION 

length is found in the elevation, or diagram of triangles. 
Point 2 at the top is distant from point 1 equal to the 
length of one space of the circle which is the true form of 
the small collar at the top, as shown in Fig. 111. Point 2 
at the base of the pattern is distant from point 1 equal 
to the distance between points 1 and 2 of the large semi- 
circle in plan. The true distance from point 1 at the 
base of the pattern to point 2 at the top is found in the 
diagram of triangles. Points 3 of the pattern are located 
in the same general manner, using measurements as 
found in Fig. 111. 

It will be noted that the intersecting plane A Y has 
produced points G F E D C B and A, therefore we must 
refer to the section A Y to secure the true distance be- 
tween said points. It may be explained that the flat 
triangular piece spoken of in Chapter 25, shows itself 
in this example at 5 F E. 

Having developed the semi-pattern as shown at Fig. 
112, it may be revolved upon line 1 1 and duplicated to 
complete the pattern for one prong, in this instance of a 
four pronged fitting. As has been previously explained, 
the position of the intersecting plane which is presumed 
to cut the original form, at once determines the number 
of prongs in the fitting when all are equal. 

Unequal Prongs. 

We sometimes hear discussions on branched fittings 
with three or more prongs which are unequal. This at 
once complicates the work of developing the pattern, 
although we may proceed along similar lines. Having 
secured a form for the fitting at the junction of its prongs 
different formed prongs may be introduced. That is, 
prongs of different diameters and radiating at different 



THE BRANCHED FITTING 231 

angles. This involves a pattern for each prong. The 
only thing' in common is the form at the junction of the 
prongs. Examples of this class arc usually more in the 
nature of a stunt than a necessity. 



CHAPTER XXVII. 

The Right or the Scalene Cone Considered in 
Securing the Patterns for a Branched Fitting. 

In some instances very satisfactory results may be 
secured by placing one or more branches at the base of 
the frustum of a right or a scalene cone when a branched 
fitting is demanded. 

The requirements of the fitting at once determine the 




Fig. 113. A Three Pronged Fitting Where the Conical 
Form is Employed in the Main Stem. 

class of a cone which should be employed. If the fitting 
is to be what is commonly known as "on center," the 
right cone is employed, and if "off center" or "flat on 
one side," the scalene cone is utilized as the main stem. 
Fig. 113 represents a three pronged fitting whose pat- 

232 



Till-: CONE CONSIDERED 



233 



terns have been secured by utilizing the frustum of a 
cone as the main stem. 

Properties of the Right Cone. 

Some properties of the right cone will be explained 
before taking up the actual pattern demonstration, since 
an understanding of the cone is essential to a successful 
handling of the problem in hand. 




Fig. 114. An Elevation of a Right Cone 

and Lines of Section. 

The cone is denned as an object which tapers uni- 
formly from a circular base to a point. If the point lies 
in the perpendicular from the center of the base, the 
cone is a right cone, otherwise an oblique or a scalene 
cone. The right cone may be conceived as being gen- 
erated by the revolution of a right angle triangle about 
its perpendicular, therefore may be looked upon as a 
solid bounded by a conical surface, and a plane which 



234 TRIANGULATION 

cuts all rectilinear elements of the conical surface. This 
plane is called its base, and the perpendicular distance 
from the plane of the base to the vertex is its altitude. A 
line, real or imaginary, from the center of the base to the 
vertex is its axis. 

In most cases the cone is looked upon as having a 
circular base, however, this will not always follow, since 
the base may have a variety of forms. In this demon- 
stration it will be assumed that the cone has a circular 
base. 

The cone with a circular base can be regarded as a 
pyramid of an infinite number of faces, hence the cone 
has in general the properties of a pyramid. The true 
form of section which will be referred to in the following 
discussion is the form which the cone would present to 
the intersecting plane when viewed at right angles to it. 

Forms Secured from the Right Cone. 

Forms secured by cutting the right cone with a cir- 
cular base are known as conic sections. Fig. 114 has 
been included for the purpose of conveying to the reader 
through the medium of the eye, a clear understanding of 
the positions of cutting planes as spoken of in the fol- 
lowing definitions. 

Definitions. 

1. If a cone be cut by a plane which passes through 
its vertex and base, as D E, Fig. 114, and making any 
angle with those parts, the true form of its section is an 
Isosceles Triangle. 

2. If a cone be cut by a plane which is parallel to its 
base as F G, Fig. 114, the true form of its section is a 
Circle. 



THE CONE CONSIDERED 235 

3. If a cone be cut by a plane which passes through 
its opposite sides, but not parallel to its base, as H J, Fig'. 

1 14, the true form of its section is an Ellipse. 

4. If a cone be cut by a plane parallel to its axis, but 
not through it, as A' L, Fig. 114, its true form of section 
is a liyperbola. 

5. If a cone be cut by a plane parallel to one of its 
sides as M N, Fig. 114, the true form of its section is a 
Parabola. 

The projection of a cone, the axis of which is perpen- 
dicular to the plane of projection, will be a circle, the 
diameter of which will be equal to the diameter of its 
base; therefore if a circle be drawn whose diameter is 
equal to the length of line A C, Fig. 114, said circle may 
be looked upon as a plan of the cone whose elevation is 
A B C. 

Some Simple Principles Explained. 

Since the representations of conic sections depends 
upon our ability to represent a given point in plan and 
elevation, which is presumed to be upon the surface of 
the cone, some of the more simple principles will be ex- 
plained. 

For an example, we shall presume that A B C, Fig. 

1 15, is an elevation of a cone, and that D is a given point 
upon its surface. To locate point D in plan we must 
first have a plan of the cone, as shown in the circle E F, 
when we may draw a line from B, the vertex of the cone 
in elevation, through point D, and intersecting the base 
line in point G. This line then becomes a rectilinear 
element of the conical surface, the plan of which may be 
secured by letting fall a vertical projector from point G 
to intersect the circle in point K. A line drawn as K H 



236 TRIANGULATION 

then becomes a plan of the rectilinear element whose 
elevation is line B G. A vertical projector dropped from 
point D to intersect line K H, locates the plan of point 
D as at M. 

*5 



Fig. 115. The Plan and Elevation of a Right Cone 
With Points Presumed to Be Upon Its Surface. 

Had the given point D been located at or near the 
center of the cone in elevation as at AT, its location in 
plan could not have been found as above described. 
Therefore it may at times be more desirable to adapt that 



THE CONK CONSIDERED 



237 



method which is the most general in application, al- 
though it has a greater tendency toward confusion. On 
the other hand, if we comprehend each method, we may 




Fig. 116. The Plan and Elevation of a Right 
Cone and Its Trite Form of Section. 



use that which best suits our purpose, therefore it is ex- 
plained below. 

Let A B C, Fig. 1 16, represent the elevation of a cone, 
and the circle D E its plan. If F is the given point in 



238 TRIANGULATION 

elevation, we may find its location in plan as follows : 
Through point F and parallel to the base line of the cone 
in elevation, draw a line as G H. From points G and H 
drop vertical projectors to intersect the line D E in plan, 
as at K and L. Upon drawing a circle as K L, we have 
before us the true form of the cone if cut by a plane 
whose elevation is the line G H. A vertical projector 
dropped from point F to intersect the circle K L as at M, 
locates point F in plan. Should it become desirable to 
determine the distance through the cone at point F, or 
the length of a line whose end elevation is at F , the 
length of line M N will supply it, presuming said line 
terminates at the surface of the cone at each side. Upon 
a moment's reflection, it will be readily understood how 
these methods may be reversed to locate points in the 
elevation whose positions are given in plan, providing 
the vertical hight of the cone is known. 

The True Form of the Section of a Cone. 

Methods as explained above may be employed to se- 
cure the true form of the section of a cone, presuming 
said cone to be cut by a plane whose elevation is a given 
line. For example, A B C, Fig. 117, is an elevation, and 
the circle directly beneath it is the plan of a cone. If it 
be presumed that the true form of section of this cone 
was required when cut by a plane whose elevation is the 
line A D, we would proceed somewhat as follows : 
Divide the circle into a number of equal parts as shown, 
and draw lines from said points of division to the center 
E. Thus we have before us the plan and elevation of 
the cone, together with the plans of a number of elements 
of the conical surface. The elevations of said elements 
are secured by projecting points of the circle as 2 3 4, 
etc., to the base line of the cone in elevation. From 



THE CONE CONSIDERED 239 

points thus located along the line A C we draw lines to 
the vertex B as shown, thereby locating points as a b c d, 
etc., upon the surface of the cone in elevation. Said 
points, i.e., a b c d, etc., may now he looked upon as the 
end elevations of lines whose extremities are at the sur- 
face of the cone. 

In any convenient position draw a line as G H, which 
is parallel to A B. From points along line A D project 
lines as shown at a b c d, etc. As will be noted, these 
lines are at a distance from each other along line G H, 
equal to the distance between points along line A. D. 
Thus we have only to determine the length of these lines 
to locate points through which a line may be traced to 
show the true form of the cone when cut by a plane 
whose edge view is line A D. These lengths are de- 
termined in precisely the same manner as has been ex- 
plained, and shown at Figs. 115 and 116. If no difficulty 
has been experienced in determining the true length of a 
single line presumed to pass through the cone from a 
given point, no difficulty should be experienced here, 
since this is simply a number of such examples. 

We may, for an example, select point a in elevation, 
where, as will be noted, a vertical projector is dropped 
to intersect line 2 E in plan, thereby locating a point in 
plan as Y, whose elevation is a. Since the vertical pro- 
jector from point 2 passes through point 16, the element 
E 16 must be directly back of element 2 E when shown 
in elevation. Therefore we may look upon the line a B 
in elevation, as not only the elevation of an element 
whose plan is 2 E, but of a similar element whose plan 
is 16 E. This applies in all cases where a vertical pro- 
jector intersects two original points of division of the 
circle in plan. Thus the vertical projector from point a 
supplies the length of the line whose elevation is a in 



240 



TRIANGULATION 



the length of said line between its intersections with 
lines 2 E and 16 E, as shown at X and Y. 

The distance from on the line 1 E to X or Y set off 
from line G H on the line projected from a, supplies the 




Fig. 117. The Plan ami Elevation of a Right Lone and the 
True Form of an Oblique Section. 

location of point a of the elevation as well as a similar 
point on the opposite side of the cone. 

A similar line of reasoning will enable one to locate 



THE CONE CONSIDERED 241 

points on the side of the cone nearest the eye, as beef 
and g, together with similar points beyond. Point (/ 
cannot be located in this manner; however, since the true 
form of the cone on a plane whose elevation is a hori- 
zontal line drawn through point (/ as S T, we note that 
the distance to be set off on each side of line G H is that 
found at d S or (/ T. 

Some attention to the foregoing should enable the 
student to find the true form of section of a cone when 
cut by a plane, regardless of the direction of said plane, 
providing its location is so taken that its elevation may 
be represented by a single line, which denotes that said 
plane is perpendicular to the vertical plane of projection, 
and at an angle other than a right angle to the horizon- 
tal plane only. 

A Common Error. 

The student is here cautioned against falling into an 
error which has many times come to the author's notice, 
and arises from a lack of knowledge of the relative posi- 
tions of the object and planes of projection. The author 
has noted instances where the operator was utterly un- 
able to solve his problem for the above reason. It seems 
a simple matter for one to conceive the object in such 
positions as to complicate the problem beyond its solu- 
tion. It should be remembered that our diagrams must 
be so constructed in examples of this nature, or at least 
the planes so taken, as to allow the intersecting plane to 
be perpendicular to one plane of projection. If the 
original views do not supply this, either planes or the 
object must be revolved until the above conditions are 
fulfilled. 

In the foregoing considerable space has been devoted 
to the more elementary work which may be involved 



242 



TRIANGULATION 



when developing the pattern for a fitting as illustrated 
at Fig. 113. In the following, our work will be confined 
to a discussion of methods which may be pursued to se- 
cure the patterns for the main stem and one branch, 
since the second branch is but a duplicate, although 
formed in the opposite direction. 

Fig. 118 is a diagram which may be looked upon as 
an elevation, or a section of a two pronged fitting, where- 
in the main stem may be considered as a portion of a 
right cone. For example, the diagram B F E D C rep- 




Fig. 118. The Elevation or Section of a Two Pronged Fitting. 

resents the frustum of a right cone, and the diagram 
A G F H B is an irregular form which constitutes the 
branch, the two forms being connected on line F H. 
With this in mind, we note that the length of line A G 
represents the diameter of the collar on the branch, and 
the lengths of lines E D and B C represent the diameters 
of the collars or ends of the conical form. To develop 
the pattern we must have a plan and a somewhat more 
complete elevation, or at least, we must locate lines pre- 
sumed to be upon the surface of the object in elevation. 



THE CONE CONSIDERED 243 

However, since the axes of all collars are here presumed 
to be in one plane, a semi-plan fulfils every requirement. 

On the Development of the Conical Surface 
With a Slight Taper. 

Since the main stem is a conical form, its patterns 
may be developed without the aid of triangulation. On 
the other hand, conditions are frequently met with 
wherein the flare or taper of the frustum of a right cone 
is so slight that the vertex of the cone, of which the 
lower part is required, would lie far beyond a reason- 
able surface upon which our elevation is to be drawn. 
In examples of this description very satisfactory results 
can be secured by applying triangulation. Since all recti- 
linear elements or direct lines are of equal lengths, and 
all indirect or those usually shown dotted, are also of 
equal lengths, to develop the surface, it therefore only 
becomes necessary to determine the true lengths of but 
two lines which are presumed to be upon the surface of 
the conical form. 

For example, Fig. 119 shows two semi-circles which 
we may presume to be the semi-plan of the frustum of a 
right cone, and the distance 1 A is its vertical hight. 
The line 1 B is then the true length of a rectilinear ele- 
ment of the conical surface, or what we have termed, a 
direct line. The true length of the indirect line 1 2 is 
shown at 2 B. One section of the conical surface is 
shown at E, this diagram having been secured by the 
use of lines shown, as in all examples where triangula- 
tion is employed. The diagram shown at E duplicated 
in this instance 16 times, will complete the pattern for 
the conical form. Some slight inaccuracy may have de- 
veloped when a pattern has been secured in this manner, 



244 



TRIANGULATION 



although this may usually be corrected without causing 
any noticeable distortion. All direct lines upon our pat- 
tern are looked upon as rectilinear elements of the conical 
surface. 

The Cone and Pyramid Compared. 

If we choose the diagram shown at E, Fig. 119, may 
be looked upon as one side of the frustum of a pyramid 
which has as many sides as parts in the circles which 
represent the plan of the conical form. Therefore if 




Fig. 119. The Semi-Plan of the Frustum 

of a Right Cone and One Section of Its 

Surface. 

the diagram E be revolved about its long side upon the 
plane of development 16 times, that surface which has 
been covered by the templet will then very closely ap- 
proximate that of the conical form. 



THE CONE CONSIDERED 245 

Upon turning attention to Fig. 120 it will be noted 
that we have before us a semi-plan and an elevation of a 
branched fitting, with the true form of an oblique section 
of the conical portion overlapping the elevation. The 
semi-circles in plan have been divided into a number of 
equal parts as at 12 3 4, etc., and lines drawn to connect 
said points as 1 1, 2 2, 3 3, etc. These lines are the plans 
of lines presumed to be upon the surface of the conical 
form, and upon projecting points as 1 2 3, etc., to the 
lines which represent the extremities of the conical form, 
we locate points between which lines are drawn to secure 
the elevations of said lines as shown, thereby locating 
points as A B C and D. 

The curved line A B C D 5 shown in plan, is the plan 
of the line A B C D 5 shown in elevation. Points in this 
line are located by dropping vertical lines from points 
A B C D 5 in elevation to intersect similar elements of 
the conical form in plan. 

Since the work of securing the true form of section of 
the conical form when cut by a plane has been previously 
shown in this chapter, the construction lines shown in 
Fi?. 120 should be sufficient to enable the student to 
determine the true form of the object upon line A 5 of 
the elevation, as shown at the true form of section A 5. 

The semi-ellipse f g h, etc., of the plan, is a plan of 
the round collar whose true form is the profile shown. 
Lines are drawn as shown in plan and elevation to repre- 
sent lines presumed to be upon the surface of the branch. 
With these lines located as shown, the diagram of tri- 
angles may be drawn in the usual manner, as also shown. 

The Semi-Pattern for the Main Stem. 
We may develop the semi-pattern for the main stem 



246 



TRIANGULATION 




THE CONE CONSIDERED 



247 



in anv manner which secures moderately satisfactory 
results, and locate lines upon its surface which have been 
previously located in plan, as shown at Fig. 121. To 
locate the line upon which the envelope of the conical 
form should be cut away to receive the branch, we may 
draw horizontal lines from points B C and D of the ele- 
vation to intersect the line 1 1 of the elevation, as shown 
at X Y and Z. Thus we have established distances along 
the line 1 1 in points A X Y and Z at which elements of 
the conical form shall he cut. 

For example, the distance 1 A found in the elevation 
is transferred to the pattern as 1 A. The distance 1 X 
found in the elevation is transferred to line 2 2 of the 
pattern, thereby locating points A and B of the pattern. 




/•"j.C. 121. The Semi-Pattern for the Main Stem of a Fitting, 
the Plan and Elevation of Which is Shown at Fig. 120. 

Fig. 121, and so on for all points shown. A line traced 
through points thus located defines the boundaries of 
that portion of the semi-pattern for the main stem 
which is to be cut away to receive the branch. 

The semi-pattern for the main stem revolved about 



248 



TRIANGULATION 



the line 1 A or 9 9 and duplicated, then completes the 
whole pattern, which in this instance may be formed in 
either direction, since the fitting is "on center." Had 
this pattern been designed for a fitting with two prongs 
which were equal, then that portion of the semi-pattern, 
Fig. 121, shown at A 1 5 5 would be the pattern for one- 
quarter of the main stem. 

The Semi-Pattern for the Branch. 

That portion of the diagram to the left of lines ABC 
D 5 shows in plan and elevation the branch, together 
with lines presumed to be upon its surface. To de- 




Fig. 122. Semi-Pattern for the Branch, the Plan 
and Elevation of Which is Shown at Fig. 120. 

velop the pattern as shown at Fig. 122, is to place upon 
the plane of development these lines in their correct rela- 
tive positions. This is an operation which has been too 
often explained in these pages to require additional dis- 
cussion, since we have shown in the diagram of triangles 



THE CONE CONSIDERED 



249 




Fig. 123. Plan, Elevation and True Form of Section for a Two or Three 
Pronged Fitting With Que of Its Sides Tangent to One Plane, 



250 TRTANGULATION 

the true lengths of said lines. It may he explained that 
in actual work it would he unnecessary to secure the 
true form of section, since if the semi-pattern for the 
main stem is first developed, we then find the true dis- 
tance between points as A B C D and 5. 

When It Is Required That the Fitting Shall Be 
"Off Center," or Tangent to One Plane. 

Should the pattern cutter be called upon for a pattern 
of this class wherein the specification demands that said 
fitting shall be "off center," or tangent to one plane, Fig. 
123 should now offer all necessary suggestions. As will 
be noted, Fig. 123 shows a semi-plan and an elevation 
wherein the above conditions are complied with. The 
method of determining the true form of section is clearly 
shown, and beyond the fact that the whole pattern must 
be developed, it differs in no essential respect from those 
previously shown, although we are now dealing with an 
oblioue cone for the main stem. 



CONCLUDING REMARKS. 

The student in sheet metal pattern cutting who has 
followed this work will have become convinced that a 
considerable number of pattern demonstrations has been 
presented. Problems which are of more or less frequent 
occurrence in the sheet metal shop. 

It is not to be expected that a work of«this character 
can be made to embrace all subjects which may come be- 
fore the operator. However, principles and methods are 
pointed out which may be employed to secure the pat- 
terns for a far greater variety of forms than can be rea- 
sonably included in any one volume. 

Study is one important factor in securing an under- 
standing of the art of pattern cutting, and must be in- 
dulged in by those who wish to become proficient. 
Pattern cutting is a branch of science which cannot be 
acquired in a da}'. There are fundamental principles 
which must be followed to secure results. A broad un- 
derstanding of those principles reduces intricate problems 
to but simple operations. 

The author, believing that it is far better to reason 
than to remember, has in this work pointed out those 
principles involved in that branch of pattern cutting 
where triangulation is applied. 

An understanding of mechanical or geometrical 
drawing involves an understanding of the more difficult 
portions of sheet metal pattern cutting. Those who 
have had the advantage of securing that understanding, 
will find this work a series of suggestions. Those who 

251 



252 TRIANGULATION 

have been denied that advantage, will find a study of 
mechanical or geometrical drawing a material aid. 

Chapters 9 and 10 have been devoted to the funda- 
mental principles of Orthographic Projection, commonly 
known as mechanical drawing. Diligent application to 
these will secure surprising results ; however, one should 
hardly expect to find this subject exhaustively treated in 
a work on pattern cutting. 

Practically all problems which demand Triangulation 
are in close relation. At first sight some appear simple, 
while others appear complex. The complex example is 
usually one in which the work in prolonged simply for 
the reason that a pattern must be developed for each 
component part of the object for which a pattern is re- 
quired. Many times the pattern for a component part 
of a complicated form is as simple of development as 
many of the so-called simple forms. The trouble ex- 
perienced by many of our pattern cutters is due solely to 
their inability to make a proper analysis of the problem 
in hand, or to their inability to make a correct geometri- 
cal representation of the object. 

There are two qualifications necessary to make a suc- 
cessful pattern cutter: these are a good understanding 
of orthographic projection and a good power of con- 
ception. The man who possesses these will have little 
difficulty in his work. We frequently meet a man who 
has an excellent power of conception, but who has de- 
voted very little time to the study of the underlying prin- 
ciples, relying upon his ability to find an untrodden path 
by which he could secure his patterns in record time. 
This often leads him into embarrassing positions. There 
are underlying principles, an understanding of which is 
as necessary to the professional pattern cutter as the un- 



CONCLUDING REMARKS 253 

derstanding of his notes is necessary to the professional 
musician. 

He who wishes to excel in the art of sheet metal pat- 
tern cutting should devote time to the study of ortho- 
graphic projection. While this may seem useless work 
and lost time in the beginning, it will be found that the 
end amply justifies the means, since he will have placed 
himself in a far better position to simplify his problems. 
In proof of the above we may ask : Who would be better 
prepared to simplify a mathematical problem than an ex- 
pert mathematician ? 

He who attempts to develop the patterns for a variety 
of forms to be constructed of sheet metal without giving 
the underlying principles some attention is a man grop- 
ing in the dark. Many such instances have come before 
the author's notice in his long period of observation. 

The author has long been a close student of sheet 
metal pattern development, and has developed patterns 
for a great variety of forms which have been made from 
all gauges of material, from light tin plate to 3/16-inch 
} ron — patterns for fittings which could be placed in the 
coat pocket, and patterns for fittings the weight of which 
would exceed a ton. There is no difference in the prin- 
ciples involved, while no doubt a somewhat greater power 
of conception is required in the larger work, since it is a 
difficult matter to secure a sufficient surface upon which 
complete diagrams may be drawn to represent to object 
upon the required plane. 

W r e may in some instances reduce the required sur- 
face by employing a scale drawing, say one-quarter size 
or 3 inches to one foot. This requires every measure- 
ment to be multiplied four times when placed upon the 
material. Tt should be understood that any inaccuracy 
is thus magnified fourfold. 



254 TRIANGULATION 

In the lighter stock some inaccuracy may be easily 
remedied, but in the heavier material this becomes much 
more troublesome. In forms where the component parts 
are assembled by double seaming, or by the use of the 
bench machine, proper allowance can be easily deter- 
mined. In the heavier work, where holes must be 
punched in the flat and coincide when the object is as- 
sembled, considerable accuracy must be maintained, and 
the thickness of the material reckoned with in every in- 
stance. In practically all examples throughout this 
work, to avoid confusion, circles have been divided into 
sixteen parts. This is in no sense a recommendation 
for the universal use of that number as explained in 
Chapter IV. 

The subject matter included in this work represents 
an honest endeavor on the part of the writer to place be- 
fore the mechanic or student, something worthy of at- 
tention by those interested in Triangulation as Applied 
to Sheet Metal Pattern Cutting. 



GLOSSARY. 

ANGLE. — The point or line on the inner or outer 
side, where two lines or surfaces meet. In a strict 
mathematical sense it signifies that relation of lines 
which is measured by the amount of rotation necessary 
to make one coincide with the other. This amount is 
usually expressed in degrees. 

ARC. — A part of a circle. 

AXIS. — One of the principal lines through the cen- 
ter of a figure or solid, especially the longest or shortest, 
or a line as to which the figure or solid is symmetrical. 

AXES.— Plural of Axis. 

BISECT. — To divide into two equal parts. 

CENTER. — The middle point of a closed curve or 
surface; properly a point such that any straight line 
drawn through it will meet the curve or surface at equal 
distances on each side of the point. 

CHORD. — A straight line connecting the extremities 
of an arc. 

CIRCLE. — A plane figure bounded by a curved line 
called the circumference, everywhere equally distant 
from a point within called the center. 

CIRCUMFERENCE.— The boundary line of a circle, 
also of any plane figure that is bounded by a curved 
line. The boundary line of any space. 

CONCEPTION.— The act or process of forming the 

255 



256 TRTANGULATION 

idea or a notion of a thing, or the idea or notion 
formed. 

CONE. — The cone is an object which tapers uni- 
formly from a circular base to a point. If the point lies 
in the perpendicular from the center of the base the cone 
is a right cone, otherwise an oblique or a scalene cone. 

CONICAL. — Shaped like a cone ; conic. 

CONVERGE. — To trend toward one point ; to incline 
and approach nearer together; direct toward a common 
focus. 

CONVERGENT. — Tending to one point; approach- 
ing each other as they extend ; said of lines. 

CROSS-SECTION.— The section of a body at right 
angles to its length ; as the cross-section of a gas pipe. 

CUBE. — A solid bounded by six equal squares and 
having all its angles right angles. 

CURVE. — Having a different direction at every 
point. 

CURVILINEAR.— Formed by curved lines. 

CYLINDER. — A solid whose curved bounding sur- 
face is generated by the motion of a straight line, re- 
maining parallel to itself, around two equal circles in 
parallel planes, the circle forming the rest of the bound- 
ary; called right when the line is at right angles to the 
planes, oblique when it is not ; in the higher geometry, 
any curved surface generated by the motion of a straight 
line remaining parallel to itself and constantly intersect- 
ing a curve. 

DEGREE. — A unit of angular measure, the ninetieth 
part of a right angle. 



GLOSSARY 257 

DESCRIPTI VE GEOMETRY.— That application 
of geometry in which the relation of lines and figure? 
are studied on planes. 

DESIGNATE. — To cause to be known or recogniz- 
able by some mark or sign. 

DEVELOP. — To change the form of a surface by 
bending or unbending without changing its smallest 
part. 

DIAGONAL.— Extending obliquely from corner to 
corner, a straight line or plane passing from one angle 
or corner to any angle or corner not adjacent to it. 

DIAGRAM. — A figure drawn to aid in demonstrat- 
ing a geometrical proposition or to illustrate geometrical 
relations. A mechanical plan or outline. 

DIAMETER. — A line through a plane figure or 
solid, terminated at the boundary thereof ; the length of 
such a line. The term is applied mostly to circular and 
spherical figures. 

DIMENSION. — Any measurable extent or magni- 
tude, as of a line, surface, or solid. 

DUPLICATE. — To make an exact copy of; repro- 
duce exactly. 

ELEMENT. — A component or essential part, espe- 
cially a simple part of anything complex. 

ELLIPSE. — A plane curve such that the sum of the 
distances from any point of the curve to two fixed points 
(called the foci) is a constant. 

ELLIPTICAL.— Shaped like an ellipse. 

FOCUS, — The point of meeting. The central point. 



258 TRIANGULATION 

FRUSTUM. — That which is left of a solid, usually a 
cone or pyramid, after cutting off the upper part. 

GEOMETRY. — The branch of pure mathematics 
that treats of space and its relation; the science of the 
mutual relations of points, lines, angles, surfaces, and 
solids, considered as having no properties but those aris- 
ing from extension and difference of situation. 

GEOMETRICAL. — Of or pertaining to geometry; 
according to the rules or principles of geometry. 

HELICAL. — Pertaining to, shaped like, or following 
the course of a helix or spiral. 

HELICOID. — A surface resembling that of a screw; 
especially one generated by a straight line, one end of 
which moves along an axis while the other describes a 
spiral about it. 

HELIX. — A line, wire, or the like, curved into shape 
such as it would assume if wound in a single layer 
around a cylinder. 

HORIZONTAL.— In the direction or parallel to the 
horizon ; or on a level. 

HYPOTHENUSE.— The side of a right angled tri- 
angle opposite the right angle. 

INTERSECTION.— A place of crossing; the point 
where two lines or the line in which two surfaces cross 
each other. 

ISOSCELES TRIANGLE.— See Triangle. 

LINE. — A line is that which has only one dimension : 
length, a straight or right line is the shortest length be- 
tween two points; a broken line is a line composed of 
different successive straight lines. A curved line is a 



GLOSSARY 259 

line no portion of which is straight. The intersection of 
two lines is a point. 

A I ITER. — The junction of two bodies at an equally 
divided angle, a piece cut at an angle for inhering, or 
pieces so cut and joined. 

OBLIQUE. — Deviating from the perpendicular or 
from a direct line by any angle except a right angle ; not 
parallel nor at right angles ; neither perpendicular nor 
horizontal. 

OBLIQUE CONE.— See Cone. 

OCTAGON. — A plane figure with eight sides and 
eight angles. 

ORTHOGRAPHIC— Of or pertaining to right lines 
or angles; drawn or projected by right lines. See Pro- 
jection. 

PARALLEL. — Lying in a plane and not meeting no 
matter how far produced; said of equidistant straight 
lines. Lines or surfaces lying in the same direction. 

PARALLELOGRAM.— A four-sided plane figure 
whose opposite sides are parallel. 

PERPENDICULAR.— Being at right angles to the 
plane of the horizon ; straight up and down. Meeting a 
given line or surface at right angles. 

PERSPECTIVE. — Delineation of objects as they ap- 
pear to the eye. Specifically, in mathematics, a branch of 
projective geometry. 

PLAN. — A drawing showing the parts in their pro- 
portion as well as relation, as of a building or machine. 

PLANE. — A surface such that a straight line joining 



260 TRIANGULATION 

any two of its points lies wholly in the surface; more 
precisely a surface which, when turned over, is congruous 
with itself, however applied. Hence, in common use, 
any flat or uncurved surface extending uniformly in 
some one direction. 

POINT. — That which has location, but not magni- 
tude. 

POLYGON. — A closed figure bounded by straight 
lines, especially more than four; a figure having many 
angles. 

PROJECTION.— The foot of the perpendicular let 
fall from a given point to a line or plane, or the straight 
line forming the feet of perpendiculars thus let fall from 
the extremities of a straight line, more widely the figure 
on a fixed plane called the plane of projection. In Ortho- 
graphic Projection the projecting rays are parallel to 
each other. 

PROJECTOR.— That which projects. 

PYRAMID. — A solid bounded by a polygonal plane 
for its base, and by triangular planes meeting in a point 
called the vertex. 

QUADRILATERAL.— Formed or bounded by four 
lines ; four sided. 

RADIATE. — Extending or passing outward from a 
common focus. 

RADII.— Plural of radius. 

RADIUS. — A straight line from the center of a circle 
or sphere to its circumference or surface. 

RECTANGLE. — A plane quadrilateral figure having 
all its angles right angles, 



GLOSSARY 261 

RECTANGULAR.— Having one right angle, or 
more, being a rectangle. 

RECTILINEAR.— Consisting of right lines. 
RIGHT ANGLED TRIANGLE.— See Triangle. 

ROTATION.— Order of sequence. 

SCALENE CONE.— See Cone. 

SCENOGRAPHIC— The art of making drawings 
in perspective. 

SECTION. — A representation, or drawing, showing- 
something, as a building or machine, as it would appear 
if it were cut by an intersecting plane, and the portion 
between the observer and the cutting plane removed. 

SECTOR. — A part of a circle bounded by two radii 
and the arc subtended by them. 

SPIRAL. — Winding continually as on the surface of 
a cylinder, or as the thread of a screw ; helical. 

SQUARE. — A rectangle having equal sides. 

TANGENT. — Meeting a line or surface at a point 
and then leaving without intersection. 

TEMPLET. — A pattern usually flat, for shaping 
something. 

TRANSFORM.— To give a different form to; alter 
in shape. 

TRANSITION. — Change from one condition to an- 
other. 

TRANSITIONAL.— Of or pertaining to transition. 

TRIANGLE. — A figure, especially a plane figure 
bounded by three lines, called sides, and having conse- 



262 TRI ANGULATION 

quently three angles. Triangles are equilateral and equi- 
angular when all the sides and angles are equal; isosceles 
when two sides are equal and scalene when no two sides 
are equal. They are right angled when one of the angles 
is a right angle, but otherwise oblique angled. 

VERTEX. — The extreme point of a figure in a cer- 
tain direction ; especially in a triangle, the point of inter- 
section of its sides. Of a cone or pyramid, the point of 
intersection of the generating lines or bounding planes 
respectively. 

VERTICAL. — Perpendicular to the plane of the hori- 
zon, plumb, upright. 



INDEX 

PAGE. 

Additional Lines, Introduction of 180 

Advice to Obtain a Clear Understanding 33 

Answer to Interesting Question 124 

Appearance of Diagrams, Variation In 81, 83 

Application of the So-Called Rule of Thumb 158 

Application of Triangulation to Sheet Metal Pattern Cutting 3 

Avoiding Error in Transferring Circumferences 66 

Breaks and Bends 168 

Capacity of a Fitting, to Preserve 56 

Capacity of Fitting Decreased 56 

Characters to Designate Points in a Pattern Problem.... 60 

Circumference of a Circle and Its Diameter, Ratio Between 66 

Circumference of Any Circle Whose Diameter is Given, to Determine 66 

Circle and Its Diameter, Ratio Between the Circumference Of 66 

Circle, Divided Into Parts, Compared With a Polygon 12 

Circle, Dividing Into a Number of Parts 12, 103 

Circles, Division Of 132 

Common Error 241 

Comparing the Cone and Pyramid 234, 244 

Complete Plan ; When It Is Required 137 

Complex Problem, Reducing to Its Simplest Form 126 

Conception of Object 61 

Conception of Object Secured From Its Specification 25 

Cone, Projection of a 235 

Cone, True Form of the Section Of 238 

Cone and Pyramid, Comparison Of 234, 244 

Cone in Plan and Elevation, to Represent Point Upon Surface Of. . 235 

Confusion of the Novice 73 

Conic Sections 234 

Conical Surface Which Has a Slight Taper. Developing 243 

Construction of a Problem, Lines Used in the 73 

Cutting Planes 176 

Cutting Planes, Positions Of 176 

Cylindrical Surface, Elements of The 117 

Definition of a Plan 19. 259 

Definition of Triangulation 3 

263 



264 trianguLation 

PAGE. 

Developing the Conical Surface Which Has a Slight Taper 243 

Development of a Pattern ; What It Rests Upon 44 

Difference in Thickness of Material 254 

Difficulty Experienced by a Novice 25 

Distance Through a Cone From a Given Point 238 

Distortion ; When It May Be Introduced 123 

Dividing Diagrams Which Represent the Ends of the Object 38 

Division of Circles 132 

Drawing a Plan, Principles Which Govern the Work Of 34 

Drawing an Ellipse 104 

Drawing an Ellipse by Projection 102 

Drawing Outfit Required 77 

Elements of a Surface 42 

Elements of the Cylindrical Surface 117 

Elevation of a Point 71 

Ellipse 235 

Ellipse, Method of Drawing 104 

Ellipse, When to Draw 153 

Example for Practice, An Excellent 33 

Experiment, An Interesting 146 

Experiment Suggested 119 

Factor Important in Pattern Development 106 

Finding True Lengths as the Work Progresses 121 

First Angle Proj ection 72 

Fitting; When It Is to Be Made in Two Equal Halves 118 

Fitting; When It Is to Be Made in Unequal Halves 118 

Fittings, Forms Of 57 

Form Frequently Demanded 50 

Form of the Oblique Section of a Cylinder 103 

Form of the Two-Pronged Fittings at the Junction of Its Prongs.. 201 

Forms Secured From the Right Cone 234 

He Who Is Best Prepared to Simplify a Pattern Problem 253 

Helicoid 146 

Horizontal Plane of Projection 69 

Hyperbola 235 

Identical Patterns Developed From Two Sets of Diagrams 83,90 

Instances Where the Operator Was Unable to Solve His Problem . . 241 

Instructive Example 58 

Intermediate Section of a Fitting, To Provide the Form For 151 

Intersecting Line 69 



INDEX. 265 

I'AGE. 

Intersecting Surfaces 1 74 

Intersecting Surfaces, Positions Of 176 

Introducing Intersecting Surfaces 174 

Isosceles Triangles 234 

Lengths of Fittings 56 

Line in Plan or Elevation ; What It May Represent 76 

Line of Penetration 174 

Lines Demanded in Orthographic Projection 8 

Lines Used in the Construction of a Problem 73 

Locating Lines Which Divide the Surface of the Object Into 

Triangles 86, 196 

Locating Point in Space From Its Plan and Elevation 209 

Locating Points in Plan 19 

Location of Triangles Upon the Surface of an Oblique Cone 22 

Logical Deductions 28 

Mechanical Method of Securing Patterns 129 

Minimizing Distortion in a Two-Pronged Fitting 200 

Miter Line, Looked Upon as the Edge View of a Plane 117 

Miter Lines and Fittings, Positions Of 107 

Numbering Points and Lines $~ 

Oblique Cone Employed in a Two-Pronged Fitting 201,208 

Oblique Plane, Supplementary Use of The 104 

Oblique Planes Illustrated 78 

Oblique Planes, Supplementary 78 

Oblique Section of a Cylinder, Form of The 103 

One Line in Elevation : When It Becomes the Elevation of Two Lines 

Upon the Surface of the Object 140, 205, 239 

One Line in Elevation; When It Represents Two Upon the Surface 

of the Object 123 

Order of Numbering Points and Lines 32 

Orthographic Projection, Lines Demanded In 8 

Orthographic Projection, Principles Of 67 

Orthographic and Scenographic Projection Compared 43 

Parallel Form Introduced 19o 

Parallelogram, To Draw a 5 

Path of Lines Drawn Upon a Pattern 47 

Pattern Cutting, Obtaining Proficiency In 253 

Pattern Development, Necessity for Lines In 49 

Pattern Problem Simplified 49 

Pattern, Simple, Secured by the LT se of Paper 6 



266 TRIANGULATION 

PAGE. 

Pattern, Securing by a Mechanical Method 129 

Patterns, Identical, Developed From Two Sets of Diagrams 83,90 

Perspective, The Value Of 28 

Plain Triangle, To Draw a 4 

Plan and Elevation Curtailed 190 

Plan and Elevation, Relative Position of The 72 

Plan and Elevation : Where They May Be Dispensed With 144 

Plan Defined 19 

Plan Drawn From a Conception of the Object 25 

Plan Drawn to Given Dimensions 13 

Plan of a Point 71 

Plan, Where It Becomes Unnecessary 149 

Planes of Projection, Reference to The 82 

Planes of Projection, Relative Positions of The 178 

Planes of Projection, Vertical and Horizontal 69 

Points in a Pattern Problem, Characters to Designate 60 

Points in Plan, Locating 19 

Point in Plan or Elevation : What It May Represent 75 

Points, Increasing Number of to Promote Accuracy 132 

Point Upon Surface of a Cone in Plan and Elevation, To Represent. . 235 

Point Which Is Often Overlooked 125 

Popular Demand, Endeavor, Made to Satisfy 113 

Positions of the Object 83 

Position of the Object in Space 70 

Position of the Object in Space as Regards the Planes of Projection. . 71 

Positions of Cutting Planes 176 

Positions of Intersecting Surfaces 176 

Positions of Miter Lines in Fittings 107 

Positions Taken for Plans, Elevations and Sections 72 

Proficiency in Pattern Cutting, Obtaining 253 

Profile 98 

Profile Plane 77,117 

Profile Plane, Use Of 78 

Principles Involved, Necessity of a Clear Conception 18 

Principles of Orthographic Projection 67 

Problem ; Simplifying or Complicating 1 76 

Projecting Lines of a Point 71 

Projection Compared, Scenographic and Orthographic 43 

Proj ection of a Cone 235 

Projection of a Point 68 



INDEX. 267 

PAGE. 

Proof in the Form of the Developed Triangles 101 

Properties of the Right Cone 233 

Proving Our Work 185 

Qualifications to Make a Successful Pattern Cutter 252 

Ratio Between the Circumference of a Circle and Its Diameter 66 

Reducing a Complex Problem to Its Simplest Form 126 

Relative Position of the Plan and Elevation 72 

Relative Positions of the Planes of Projection 178 

Representation of a Point Upon the Vertical and Horizontal Planes. . 71 

Representation of the Object Upon the Vertical and Horizontal Planes 74 

Representations of Objects ; What They Are Composed Of 70 

Revolution of the Planes 72 

Right Angled Triangles; When They Are Shown in Plan 101 

Right Cone, Forms Secured From 234 

Right Cone, Properties Of 233 

Rule of Thumb, Application Of 158 

Scale Drawings, Use Of 253 

Scenographic and Orthographic Projection Compared 43 

Section of Cone, True Form Of 238 

Section of Fitting; When It Approximates a Semi-Ellipse 215 

Securing Patterns by a Mechanical Method 129 

Simplifying the Work 176 

Study Required to Conceive the Forms Whose Patterns May Be 

Developed 33 

Suggestions Upon Tapering Elbows 134 

Supplementary Oblique Plane, Use Of 104 

Supplementary Oblique Planes 78 

Surface Upon Which a Plan Is Drawn 19 

Surface Upon Which the Object Is Represented 68 

Tapering Elbows, Suggestions Upon 134 

Templet : Where It May Be Utilized 154 

Thickness of Material, Difference In • • • 254 

Things to Be Remembered 1'8 

Transferring the Exact Circumference of a Circle to the Pattern 66 

Triangles Involved in Triangulation 3 

Triangles Upon the Surface of an Oblique Cone, Location Of 22 

Triangulation, Definition Of 3 

Triangulation Applied to Sheet-Metal Pattern Cutting 3 

Trouble Experienced by Many Pattern Cutters 252 

True Form of the Section of a Cone 238 



268 TRIAXGULATION 

PAGE. 

True Length of a Right Line in Space 3 

Two Equal Halves of the Pattern ; When They Must Be Formed in 

Opposite Directions 206 

Two-Pronged Fitting, Form of At the Junction of Its Prongs 201 

Two-Pronged Fitting, Suitable Proportions For 200 

Undevelopable Surface, An 147, 160 

Variation in Appearance of Diagrams 81, 83 

Variation in Our Line of Reasoning 210 

Vertical and Horizontal Planes of Projection 69 

Warped Surface 146 

Whole Object ; When It Must Be Represented 61 



ELBOW PATTERNS for 
all FORMS of PIPE 



A treatise on the elbow pattern 
explaining the most simple and ac- 
curate methods for obtaining the 
patterns for elbows in all forms of 
pipe made from sheet metal; 

With Useful Mathematical Rules 
and Tables 

By F. S. KIDDER 

One of the first and most import- 
ant considerations for the sheet 
metal worker is to be possessed of a 
method for securing the patterns 
for elbows in the least possible time 
consistent with accuracy. To meet the popular demand and 
provide a means by which unnecessary expenditure of time 
and labor may be avoided, the author presents here a method 
for laying out elbows with accuracy, despatch, and 

WITHOUT RESORT TO GEOMETRICAL DISPLAY 

With the service of this handbook, the mechanic will be 
enabled to quickly produce the patterns for elbows in round 
pipe of any size, angle or number of pieces, by the simple 
employment of a pair of compasses and a straightedge. 

Size, 4% x6y 2 inches (for the pocket), 73 pages, 35 figures, cloth bound. 
Price $1.00. 




SHEET METAL PUBLICATION COMPANY 

Tribune Building, New York 



PRACTICAL SHEET METAL 
DUCT CONSTRUCTION 

A Treatise on the Construction and Erec- 
tion of Heating and Ventilating Ducts 

By WILLIAM NEUBECKER 

This is a new and exceptional 
book for sheet metal workers, none 
of whom should miss having it. It 
is full in description and illustra- 
tion of almost every operation com- 
prised in the handling and forming 
of materials and the erecting" and 
installing of ducts for heating and 
ventilating systems, from the cut- 
ting of the metal to the erection of 
the ducts in buildings. There are 
217 illustrations, mostly from pen 
and ink drawings, making plain 
every operation referred to in this 
clear and carefully prepared man- 
ual. Many of the chapters are in- 
valuable in suggestion and useful- 
ness to all sheet metal workers, 
whatever branch of the business 
they are engaged in. 

C O N T E NTS 




Chapter 

1 — Tools, Machinery and Mate- 
rials. 

2 — Sheet Metal Work in Heating 
and Ventilating Systems. 

3 — Cutting Material, Bending, 
Forming, Seaming, Grooving 
and Bracing. 

4 — Constructing the Various Slip 
Joints. 

5 — Hanging Ceiling Ducts — Sup- 
porting and Fastening Ducts. 

6 — Methods of Laying Out and 
Constructing the Elbows. 

7 — Constructing Register Boxes, 
Cleanouts and Ceiling Venti- 
lators. 

8 — Registers for Heating and 
Ventilating Dints. 



Chapter 

9 — Fastening and Adjusting 
Registers. 

10 — Connecting Heating and Ven- 
tilating Ducts. 

11 — D ucts for Direct-Indirect 
Heating. 

12 — Construction of Casings for 
Indirect Heating. 

13 — Assembling Casings and Con- 
structing Air Filters. 

14 — The Construction of Mixing 
Dampers. 

15 — Making Dampers for Farge 
Ducts. 

16 — Calculating Areas of Pipes and 
Ducts. 

17 — Various Types of Ventilators. 

18 — Construction of a Large Ven- 
tilator. 



Substantially Bound in Cloth, 191 Fages, Size 5">/2 x 8^4 Inches. 
Price $2.00. 



SHEET METAL PUBLICATION COMPANY 
Tribune Building, New York 



The New Tinsmith's Helper 

and PATTERN BOOK 



A Text Book and Working Guide for the Student, 
Apprentice and Mechanic. 

By Hall V. Williams 



The New 
Tinsmith's Helper 

and 

Pattern Book 



This is an entirely new treatment 
of the subject which displaces the 
old "Tinsmith's Helper," a book 
of one-third the size of the pres- 
ent new publication. 

The new material covers each 
phase of pattern cutting', from the 
making of seams, laps and joints, 
to conical problems and tinware, 
tin roofing, elbows, piping", gut- 
ters, leaders, cornice and skylight 
work, tin clad fire-proof doors, 
furnace fittings, etc. Information 
is 'also included on laying metal 
shingles, tile, slate, etc. 

The chapter of 92 reference tables for the sheet metal 
trades adds exceptional importance and usefulness to this 
vastly improved and always popular manual. 



Hall V. Williams 



Contents 



Chapter — 

1 — Mensuration. 

2 — Simple Geometrical Problems. 

3 — Conical Problems and Tinware. 

4 — Elbows and Piping. 

5 — Furnace Fittings. 

6 — Gutters, Leaders and Roofing. 



Chapter — 

7 — Cornice Work. 

8— Skylights. 

9 — Laps, Seams and Joints — Laying 

Slate Roofing, Making Tin-clad 

Fireproof Doors, etc. 
10 — Handy Recipes and Formulas. 
11— -Useful Tables (92). 



352 Pages, 247 Illustrations, Size 5x7 Inches, 
Bound in Cloth, Price $2.00. 

SHEET METAL PUBLICATION COMPANY, 
TRIBUNE BUILDING, NEW YORK. 



THE VENTILATION 
HAND BOOK 



The Principles and Practice of Ventilation as Applied to 

Furnace Heating; Ducts, Flues and Dampers for Gravity 

Heating; Fans and Fan Work for Ventilation and Hot 

Blast Heating 

By CHARLES L. HUBBARD 

The Hand Book is designed to 
cover the entire range of the subject 
by means of a comprehensive series 
of questions, answers and very plain 
descriptions such as every one can 
understand. The subject is treated 
progressively from its elements or 
first principles to the practical appli- 
cation of ventilation in all classes of 
buildings. The author is a well known 
heating engineer possessed of a wide 
practice and large experience. Not 
less important, he is experienced and 
thoroughly qualified as a writer on 
heating and ventilation and knows 
how to handle his subject so as to im- 
part the most essential information. 
Therefore, the Ventilation Hand 
Book is reliable and thorough, con- 
stituting a first class text book for those who are interested in 
studying or reviewing ventilation practice. 

CONTENTS 




Chapter 

1 — General Definitions, Heating and 
Ventilating Requirements, Hu- 
midity. 

2 — Systems of Ventilation, General 
Arrangement, Air Meastirement. 

3 — Cold Air Supply Ducts and 
Fans. 

4 — Warm Air Supply Ducts and 
Flues. 



Chapter 

5 — Discharge Ventilation. 

6— Stack Casings and Dampers. 

7 — Fans and Fan Drives. 

8 — Air Filters and Washers. 

9 — Ventilation of Various Types of 

Buildings. 
10 — Ventilation of Toilets and 

Chemical Hoods. 
11 — Warm-Air Furnace Heating. 



Contains 328 Questions, Answers and Descriptions on Ventilation 

Illustrated by 137 Diagrams. Size 5V S x 8 Inches. 218 Pages. 

Durably Bound in Cloth. Price $2.00. 

SHEET METAL PUBLICATION COMPANY 
Tribune Building, New York 



PRACTICAL EXHAUST AND 
BLOW PIPING 

A Treatise on the Planning and Installation of 
Fan Piping in all its Branches 

By WILLIAM H. HAYES 

^s^** 55 *^ . At the present time no depart- 

^g^^^^^^sss^^^^;, nient of sheet metal work offers a 
more profitable field of opportunity 
for sheet metal workers than ex- 
haust and blow pipe work. Every 
operator should be qualified with a 
practical knowledge of the correct 
design and installation of fan pip- 
ing systems. 

This book was written by an ex- 
pert of long and varied experience 
IIIKIiiWIi in the field, as a foreman and super- 
intendent and the treatment is 
I based on practical daily work. It 

ppP^ is reliable, easily understood and 
applied, covering the essentials of 
efficient construction very clearly. 

Contains Sixteen Sections 



III 



Each devoted to a special phase of the work, including full 
and specific information on construction, installation and costs. 



General Rules. 

Connecting: Dust Separator and 
Feeder. 

Piping: for Automatic Firing: Sys- 
tem of Boilers. 

Constructing: the Feeder Nozzle 
and Switch. 

Piping: System for a Planing: Mill. 

Pipe Connections for a Flooring: 
Machine. 

Designs for Hoods and Sweepers. 

Hoods for Special Machines. 

Proper Construction of the Sepa- 
rator. 



Efficiency of the Exhaust Fan. 
Use of the Two-Way Mixing- Valve 

and the Automatic Damper. 
Piping- a Forge Shop, including: the 

Blast and Suction Systems. 
"Don'ts" and "Don't Forgets" for 

Blow Pipe Men. 
Correspondence Relating to Special 

Problems. 
Hints on Installing an Exhaust 

System. 
Hints on Estimating the Cost of an 

Exhaust System. 



160 pages, 51 illustrations. Printed on heavy paper, durably bound in 
cloth. Price $2.00. 

SHEET METAL PUBLICATION COMPANY 
Tribune Building, New York 



TRIANGULATION 

APPLIED TO SHEET METAL PATTERN CUTTING 
By F. S. Kidder 

A Comprehensive Treatise for Cutters, 
Draftsmen, Foremen and Students; Pro- 
gressing from the simplest phases of 
the subject to the most complex prob- 
lems employed in the development of 
Sheet Metal Patterns; With practical 
solutions of numerous problems of fre- 
quent occurrence in sheet metal shops. 

This new work will be universally 
welcomed by practical pattern cutters 
and students of the subject, for it is a 
book in which the subject is treated 
systematically from its elements, with 
the principles so laid down that the 
reader may acquire a ready conception 
and skill in laying out any given class of 
work. 

It is the most important contribution 
of many years to the literature for the 
sheet metal trade. 




Contents 



Chapter — 

1 — Elementary Principles. 

2 — Simple Transitional Fitting. 

3 — The Oblique Cone. 

4 — 'A Transitional Fitting from Rec- 
tangular to Round which Makes 
an Offset. 

5 — A Twisted Transitional Fitting. 

6 — The Pattern for the Frustum of 
an Oblique Cone. 

7 — A Transitional Fitting from Ob- 
long to Round. 

8 — A Two-Pronged Fitting which Can 
be Made in One Piece. 

9 — Some Principles of Orthographic 
Projection as Applied to Triangu- 
lation. 

10 — The Representation of Objects 
Upon the Vertical, Horizontal, 
Profile and Oblique Supplementary- 
Planes of Projection. 

11 — The Pattern for a Fitting whose 
Ends are not in Parallel Planes. 

12 — The Pattern for a Fitting whose 
Ends are not in Parallel Planes — 
Second Demonstration. 

13 — A Transitional Elbow from Round 
to Rectangular. 



Chapter — 

14 — A Transitional Offset from Round 

to Rectangular. 
15 — A Three-Pieced Tapering Elbow. 
16 — The Ship's Ventilator. 
17 — On the Tapering Elbow to be 

Made in Any Number of Pieces. 
18 — A Transitional Elbow in Rec- 
tangular Pipe. 
19 — A Transitional Elbow from Round 

to Elliptical. 
20 — The Helical Elbow. 
21 — When it is Required that a Round 

Pipe Should Join the Frustum of 

an Oblique Cone. 
22 — A Branched Fitting Commonly 

known as "Breeches." 
23 — A Simple Two-Pronged Fitting. 
24 — A Two-Pronged Fitting Whose 

Prongs are of Unequal Diameters. 
25 — On the Two-Pronged Fitting 

When it is Required that the 

Prongs Radiate at a Given Angle 

to the Main Stem. 
26 — On a Fitting with Any Number 

of Prongs, Glossary, Index. 



274 Pages, 144 Illustrations, Size 6x9 Inches, 
Cloth Bound, Price $2.50. 

Sheet Metal Publication Company, Tribune Bldg., New York 



PROGRESSIVE FURNACE 
HEATING 

A Practical Manual of Designing, Estimating and Installing 

Modern Systems for Heating and Ventilating 

Buildirgs with Warm Air 

By ALFRED G. KING 

Together with a Complete Treatise on the Construction and 
Patterns of Furnace Fittings 

By WILLIAM NEUBECKER 

This work constitutes a complete 
modern guide to warm air furnace 
heating, its design and installation, 
with working plans and methods of 
estimating and full information re- 
garding up-to-date accessories and 
equipment. The most approved 
practice is brought out and made 
clear, without departure from the 
uses of plain arithmetic in the mak- 
ing of calculations. An important 
feature of the work is the complete 
treatise on the construction of sheet 
metal furnace fittings, with detailed 
pattern layouts and methods for the 
operator. 

CONTENTS 






Chapter 

1 — The Chimney Flue. 

2 — The Furnace — Character, Size, 
location and Setting. 

3 — Pipes, Fitting's and Registers. 

4 — Installation of the Fiirnace. 

5 — Trunk Line and Kan-Blast Hot 
Air Heating. 

6 — Estimating Furnace Work. 

7 — Intelligent Application of 
Heating Kules. 

8 — Practical Methods of Construc- 
tion. 

9— What Constitutes Good Fur- 
nace Work. 
10 — Ventilation, 

11 — Ventilation by the Use of Pro- 
peller Fan. 



Chapter 

12 — Humidity and the Value of Air 
Moistening. 

13 — Recirculation of Air in Fur- 
nace Heating. 

14 — Auxiliary Heating from Fur- 
naces. 

15 — Temperature Regulation and 
Fuel Saving Devices. 

16 — Fuel: Its Chemical Compo- 
nents and Combustion. 

17 — Cement Construction for Fur- 
nacemen. 

18 — Construction and Patterns of 
Furnace Fittings. 

19 — Rules, Tables and Information. 

20 — Recipes and Miscellaneous Data 
Index to Furnace Heating. 
Index to Furnace Fittings. 



Size 6x9 inches, 280 pages, 189 illustrations in line (from pen and ink 

drawings) and half tone (from photographs). 

Elegantly Bound in Cloth. Price $2.50. 

SHEET METAL PUBLICATION COMPANY 
Tribune Building, New York 



KINKS 

and LABOR SAVING METHODS 
FOR SHEET METAL WORKERS 

Practical Money-making Ideas, Devices and Descrip- 
tions for the Accommodation of the Workman. 

Volume No. 1 In Press. 

Edited by J. T. Bliss and William Neubecker. 



The Kink Books, now in prepara- 
tion, will contain a rare collection of 
original methods and devices for 
shortening labor and economizing 
material, in the shop and on the job. 
There are hundreds of ideas and ex- 
pedients, all contributed by sheet 
metal workers throughout the coun- 
try, illustrated by means of cuts 
made from original drawings. 

Various methods devised and suc- 
cessfully used by mechanics, fore- 
men and superintendents, for facili- 
tating work and saving time and ma- 
terial are featured. 



No reader can fail to find a large number of time and 
money-saving methods that are new to him, any of 
which may prove a big factor for efficiency and con- 
venience in handling stubborn problems. 

Three Volumes, Each Containing About 125 Pages, 
Fully Illustrated. Size 45^ x 7 Inches. 

Price, $1.00 per Volume. 

SHEET METAL PUBLICATION COMPANY, 
TRIBUNE BUILDING, NEW YORK. 




Practical 

Sheet Metal Work 

and Demonstrated Patterns 

A Series of Companion Volumes Covering the Various 
Branches of Sheet Metal Work. 

Edited By J. H. Teschmacher. 




These twelve volumes provide the 
solutions of innumerable problems 
in shop and outside practice. They 
give the methods of laying out, cut- 
ting, forming up and erecting all 
classes of work, with valuable hints 
by experts. No ambitious mechanic 
or well-equipped shop can afford to 
be without the set, or at least select- 
ed volumes, which may be procured 
separately. 



Volume 

1 — 'Leaders and Leader Heads — 113 
pp., ISO figures. 

2 — Gutters and Roof Outlets — 116 pp., 
194 figures. 

3 — Roofing — 138 pp., 207 figures. 

4 — Ridging and Corrugated Iron Work 
— 132 pp., 239 figures. 

5 — Cornice Patterns — 119 pp., 195 fig- 
ures. 

6 — Circular Cornice Work — 126 pp., 
194 figures. 



Volume 

7 — Practical Cornice Work — 139 pp., 

237 figures. 
8 — Skylights— 122 pp., 260 figures. 
9 — Furnace and Tin Shop Work — 145 

pp., 239 figures. 
10 — Piping and Heavy Metal Work — 

144 pp., 259 figures. 
11 — Automobile and Sheet Metal Boats 

— 137 pp., 193 figures. 
12 — Special Problems — 144 pp., 150 

figures. 



Size of Volumes 8^x11 Inches, Bound in Cloth. 

Complete Set (12 Volumes) $15.00 

Any Six (6) Volumes 8.00 

Single Volumes 1-50 

SHEET METAL PUBLICATION COMPANY, 
TRIBUNE BUILDING, NEW YORK. 



HOW TO MAKE THE 
BUSINESS PAY 

A Guide for Sheet Metal Con- 
tractors on Business Manage- 
ment, Bookkeeping and Cost 
Accounting with Systems, 
Methods and Forms; Estimat- 
ing ; Correspondence and Adver- 
tising; Methods of Securing and 
Retaining Customers, Etc. 

By EDWIN L. SEABROOK 

Secretary of the National Association 

of Sheet Metal Contractors 

A glance over the Table of Con- 
tents of this new book will insure its 
purchase by every progressive Con- 
tractor in the Sheet Metal Trade, for 
there is nothing like it in print. 
Every phase of the executive handling 
contracting business has been diligently investigated by 
the author. He has drawn, not alone on his personal business 
experience, wide as it has been, but he has had access to the 
methods and systems employed by over 2,000 contractors with 
whom he had come into contact in his twelve years of experi- 
ence as an association secretary. Thus the book presents pro- 
gressive methods such as have been tested in actual practice. 

Chapter CONTENTS^ 

1 — Business Principles and Practices. 

2 — Ascertaining the true Cost of Conducting Business. 

3 — Estimating Labor and Material. 

4 — Bookkeeping. A Complete System, with Methods and 

Forms. 
5 — Effective Collection Methods. 
6 — Plans for Extending Business by Publicity. 
7 — Correspondence and How to Utilize It for Profit. 
8 — Securing and Retaining Custom. 
9 — Office Equipment and Its Uses. 

175 Pages. 6x9 in. Fully Illustrated. Cloth Covers. Price $2.00 

SHEET METAL PUBLICATION COMPANY 
Tribune Building, New York 




MENSURATION 



\Ji 



Its Application in Working Ordinary Problems of 
Shop Practice. 

By William Neubecker. 

This book explains the principles 
f mensuration as applied in meas- 
I uring" sheet metal work, taking as 
examples the various shapes of 
formed sheet metal products. The 
I purpose of the book is to show how 
to obtain the area and capacity of 
I any article of sheet metal, so that 
the pattern cutter or mechanic may 
accurately measure the material re- 
quired for a given class of work. 
The style of treatment is very simple, and furnishes 
a quick reference for every-day use. 



Contents. 



Length. 

Area. 

Convex Surfaces. 

Solid Contents. 

Prismoidal Formulas. 

Practical Examples for the Shop. 

Ascertaining the Sizes of Articles 

Short Rules in Computation for Prac- Roof Measurements. 

tical Shop Work. Measuring a Copper Cupola. 



Problems Solved by the Steel Square. 
Finding Similar Areas in Pipe Work. 
Computation in Skylight Work. 
Obtaining Radius by Computation. 
Computing the Pitch of Gutters. 
Problems Solved by the Compass and 
Steel Square. 



128 Pages, 115 Illustrations, Size 5x8 Inches, 
Cloth Bound, Price 75c. 



SHEET METAL PUBLICATION COMPANY, 
TRIBUNE BUILDING, NEW YORK. 



BEST BOOKS for 
SHEET METAL WORKERS 



WE CARRY A STOCK OF ALL THE 
BEST BOOKS FOR THE SHEET METAL 
TRADE, AND SUPPLY THEM, DELIVERY 
EXPENSES PAID. 

OUR DESCRIPTIVE BOOK CATA- 
LOGUE WILL BE MAILED ON REQUEST, 
AND INQUIRIES WILL BE ANSWERED 
PROMPTLY, REGARDING BOOKS WHICH 
CONSTITUTE RELIABLE SOURCES OF 
INFORMATION ON SUBJECTS OF IN- 
TEREST TO SHEET METAL WORKERS. 



f WM9L 2 

*TV J^m\A/ml //m\\ WV /«■* ^^ 



TRIBUNE BUILDING, 
NEW YORK. 



